School Science Lessons
25. Wave motion, transverse, longitudinal, diffraction, interference, dispersion, reflection, refraction, ripple tank, water waves, springs
2014-05-31
Please send comments to: J.Elfick@uq.edu.au

Table of contents
25.1.0 Waves
25.1.0 Ripple tank, waves, (Experiments)
25.5.0 Springs, slinky springs, (Experiments)
25.2.0 Wave motion
25.5.0 Springs, slinky springs
Sound and waves, (slinky spring) "Prof Bunsen", (commercial website)
Waves, (slinky spring), "Scientrific", (commercial website)
Experiments
16.2.1.16 Drop a slinky spring
25.2.4 Interference of waves in springs
25.2.2 Longitudinal wave with a coiled spring, slinky spring
25.2.3 Reflected wave in springs
25.2.1 Transverse wave with a coiled spring, slinky spring, pulse
25.1.1 Transverse progressive wave, (See: 2.)
25.2.5 Wavelength, frequency and speed of a wave in a spring

25.1.0 Ripple tank, (Experiments)
Waves, (ripple tank), "Scientrific",(commercial website)
See
: Waves and sound, School of Physics, University of New South Wales. [Animation needs Flash 8 Plugin.]
Experiments
25.3.1.0 Ripple tank, wave tank
25.3.3.1 Reflection of circular wave at plane barrier
25.3.7 Diffraction at narrow openings in barriers
4.92 Diffraction in a ripple tank
27.33 Diffraction in a ripple tank
28.8.4 Convex lens, concave lens
4.86 Make a ripple tank
25.3.5 Reflection of waves at a curved barrier, (curved block)
25.3.4 Reflection of waves at a straight barrier, (straight block)
25.3.6 Refraction of waves
26.6.13 Ripple tank acoustics
25.3.3 Simple circular pulses, ring-shaped wave
25.3.2 Simple straight line pulses, straight wave
25.3.1.2 Straight barrier (straight block, plane barrier, block boards)
28.5.2 Refraction tank, aquarium
26.6.13 Ripple tank acoustics, (sound)

25.2.0 Wave motion
25.00 Waves, wave motion
25.12 Annulment of waves
25.13 Beats
25.06 Huygen's principle
25.07 Laws of reflection of waves
25.08 Laws of refraction of waves
25.05 Longitudinal progressive waves
25.02 Progressive waves
25.09 Radiant energy waves
25.01 Simple harmonic waves and vibrations
25.10 Speed of wave propagation
25.1.3 Stationary wave, standing wave, with a rope or string
25.04 Wave fronts
25.03 Wave vibration phase
Experiments
26.4.13 Chladni vibrating plate
25.3.7 Diffraction at narrow openings in barriers
25.14 Doppler effect, change in pitch
26.4.12 Kundt's tube, (standing waves)
25.1.2 Reflected wave with a rope
4.90 Reflection at a curved barrier
4.89 Reflection at a straight barrier
25.3.5 Reflection of waves at a curved barrier, curved block
25.3.4 Reflection of waves at a straight barrier, straight block
25.3.6 Refraction of waves
4.91 Refraction of waves
25.1.4 Size of the slit with a rope
25.1.3 Stationary wave, standing wave, with a rope or string
4.88 Straight pulses
25.1.1 Transverse progressive wave
4.93 Sea wave patterns and sound wave patterns
4.94 Wave patterns of a tuning fork
25.1.1 Transverse progressive wave
4.85 Waves travel along a rope
25.3.1.3 Make waves
3.8 Water waves (Primary)

25.00 Waves, wave motion
See diagram 25.00: Wave motion
Waveform is the shape of a wave at any moment. It is often used to characterize the particular shape of a wave at a particular voltage
Wave motion is a natural phenomenon. You may call a wave any physical quantity that has the same relationship to some independent variable, usually time, that a propagated disturbance has, at a particular instant, with respective to space. Mechanical waves allow transport of momentum and energy through matter by the motion of a disturbance of it without any bulk motion. The matter is called the medium. The medium is elastic, continuous and extended and is not permanently displaced by the passage of a wave. Electromagnetic waves are oscillating electric and magnetic fields travelling together through space at a speed of nearly 300 000 km per second. A medium is not necessary for electromagnetic waves or gravitational waves. A wave allows the transfer of energy without any transfer of mass. It is a self sustaining travelling disturbance of a medium carrying energy and momentum. Mechanical waves are aggregate motion from the motion of constituent particles. The wave advances, but the particles of the medium oscillate about a central point.

25.01 Simple harmonic waves and vibrations
A source that oscillates with simple harmonic motion produces the simplest periodic wave called a simple harmonic wave. Complex waves are the result of simple harmonic waves. Free vibrations are the natural vibrations that occur when you disturb an elastic body, e.g. a tuning fork. Damped vibrations have amplitudes that diminish with time due to energy dissipation, e.g. a bouncing ball. Forced vibrations occur when a medium receives an applied periodic vibratory force, e.g. the movement of sand in a shaken sieve. Resonance occurs when a periodic applied force produces forced vibrations whose frequency is the same as the natural frequency of free vibration of the medium, causing large amplitude of vibration.

25.02 Progressive waves
See diagram 25.00: Wave motion
When a particle executes SHM, you can plot the displacement against time, to get a sine curve
1. Amplitude, symbol a, is the maximum displacement. It is the maximum disturbance during a vibration cycle, measured along the y axis, so it is half the peak-to-peak value.
2. Period, symbol P, is the time for one complete vibration or the time taken to complete one cycle, i.e. the time taken for a particle to move through one complete vibration down and back. Measure period in the number of cycles per second. T = 1 / f, where f is the frequency of oscillation. Period is the reciprocal of frequency.
3. Frequency, symbol f (formerly n), is the number of vibrations per second or the number of cycles in unit time, f = 1 / T. If T is in seconds, then f is in hertz, Hz, where 1 Hz = 1 cycle per second. The period and frequency of the wave are the same as the period and frequency of the original vibration.
1 kilohertz, kHZ = 1000 Hz, 103 Hz
1 megahertz, MHz = 1000 kHz = 106 Hz
1 gigahertz. GHz = 1000 mHz = 109 Hz
1 terahertz, THz = 1000 GHz = 1012 Hz
AC signals in the audiofrequency, AF, range have frequencies between 20 Hz and 20 kHz , frequencies a person with good hearing can hear when AC waves are transformed into sound waves. AC signals in the radiofrequency, Rf range, have frequencies in the whole range of frequencies.
The AC wave is a sine wave, it is sinusoidal in shape as seen on an oscilloscope. It shows time on the horizontal axis and amplitude (voltage) on the vertical axis. An oscilloscope has a time-domain display.
4. Wavelength, symbol Greek letter λ, 1. is the distance between any two successive points in the same phase, e.g., between two crests or the distance along the direction of propagation, × axis, between corresponding points on the wave. The top points on the wave are called the wave crests. The bottom points are called the troughs. Assume the crests and troughs move to the right with speed v, i.e. the speed of the wave. Measure from crest to crest because these points are easy to identify. In time T, a crest moving with speed v will move a distance of one wavelength to the right.
5. A pulse is a single wave disturbance from a single vibration source.

25.03 Wave vibration phase
The phase of the vibration at a point at any instant is specified by the following:
1. The angle turned through by the radius drawn to the corresponding point in the circle of reference, i.e. Greek letter θ. In the diagram P = π / 4 radians, 45oC.
2. The fraction of the period completed
3. The fraction of a wavelength travelled, e.g. for point P, λ / 8.
In-phase vibration exist at two points on a wave if those points are moving in the same direction, i.e. moving up together or moving down together. Particles in a vibration are in-phase if they are a whole number of wavelengths apart. Particle moving in opposite directions are said to be 180o, or half a cycle, out-of-phase. A periodic wave is a succession of wave shapes from a vibration source with repeating regular action. The periodic wave has the same frequency as the vibrating source.

25.04 Wave fronts
See diagram 25.04: Wave fronts
For a wave system, e.g. the system of waves arising from a point source, a wave front is a surface drawn through points that are all in the same phase at a given instant. For waves travelling out with equal velocity in all directions from a point source, the wave fronts are spheres, circles for a plane section as shown. Diagram 25.04.1 shows portions of successive wave fronts drawn through successive points varying in phase from that at the origin by 0, 2 π, 4πi.
Experiment
Draw the wave fronts at intervals of a wavelength. Diagram 25.04.2 shows a plane wave that has wave fronts that are planes. A spherical wave approximates to a plane wave at enough distance from the source. A ray is any line drawn perpendicular to a wave front.

25.05 Longitudinal progressive wave
See diagram 25.05: Longitudinal progressive wave
A longitudinal progressive wave, compression wave, has vibration direction parallel to the direction of propagation, e.g. sound wave. In a simple harmonic longitudinal progressive wave each particle executes an SHM in the same line as the direction of propagation of the wave. The amplitude is the same for all particles, but the phase alters progressively from particle to particle. You can represent the wave with a sine curve using the convention that you plot the displacement at right angles to the direction of propagation, i.e. at right angles to the actual displacement. Draw displacement to the right above the axis and displacement to the left below the axis. A longitudinal wave consists of a series of condensations and rarefactions, the distance between two successive condensations, i.e. points in the same phase are equal to λ, the wavelength.
Diagram A denotes the actual arrangement of some of the particles in a longitudinal wave at any instant. 1, 2, 3 round dots denote the equilibrium positions for the particles, and 1, 2, 3 crosses denote the displacement position at that instant.
Diagram B shows the corresponding conventional representation as a sine curve. The displacements can be drawn to any convenient scale. In A the displaced particles are shown in a separate line when actually the displacements are along the line of the particles themselves. The amplitude is the maximum longitudinal displacement of a particle from its undisplaced position, here nearly equal to the undisturbed distance apart of the particles examined. The arrows show the direction of displacement. Between 4 and 6 the particles are crowded together forming condensations a wavelength apart. Between 6 and 12 the particles are apart forming a rarefaction.

25.06 Huygen's principle
Huygen's principle of secondary wavelets states that any point on a wave front may be regarded as the source of secondary wavelets, which combine to form the new wave front. The new wave front is a surface tangential to, i.e., the envelope of, all the secondary wave fronts.
Diagram 25.06.1, Huygen's principle, shows the disturbance starting at the centre and represents a spherical wave from a point origin expanding with uniform velocity in all directions. A is a wave front, every point on which has then acted as a centre of disturbance, so that the new wave front B has been produced. The circles represent a section of the spheres in the plane of the paper.
Diagram 25.06.2 shows a plane wave, to which the spherical wave approximates over a small area (in every direction) at a sufficiently great distance from its origin. The results would be different if the wave velocity were different at different places in the medium, or if the velocity were not the same in all directions.

25.07 Laws of reflection of waves
When waves are reflected at a surface:
1. the incident ray, the reflected ray, and the normal at the point of incidence are in the one plane, and,
2. the angle of incidence is equal to the angle of reflection.
Diagram 25.07.1, Laws of reflection, shows the normal is the perpendicular to the reflecting surface.
Diagram 25.07.2 shows that the normal to the surface of a sphere at any point is its radius to that point. The angles of incidence and reflection are the angles between the incident ray and the normal, and between the reflected ray and the normal.

25.08 Laws of refraction of waves
When waves pass from one medium to another:
1. the incident ray, the refracted ray, and the normal at the point of incidence are in one plane, and,
2. the sine of the angle of incidence is proportional to the sine of the angle of refraction.
The fraction sin i / sine r = velocity of the disturbance in the first medium / velocity of the disturbance in the second medium = a constant, the index of refraction (refractive index).
Diagram 25.08.1 and 25.08.2, Laws of refraction, show light waves, where the velocity in glass < the velocity in air. Light travels about half as fast again in air as it does in glass, sin i / sin r = approx. 3 / 2. Sound travels about half as fast again in glass as in air, sin i / sin r = approx. 2 / 3.

25.09 Radiant energy waves
Radiant energy waves include radio waves, heat, infrared, light, ultraviolet, X-ray and y rays, and cosmic rays. All are transverse waves with velocity in a vacuum = 3 × 1010 cm. sec-1. These waves are all electro-magnetic waves. but differ in wavelength. Sound consists of longitudinal vibrations in a material medium, the velocity in air = approx. 33 000 cm. sec-1 (1 100 ft. sec.-1) Only a limited range of wavelengths are audible but longer and shorter waves can be detected by equipment.

25.1.1 Transverse progressive wave
See diagram 25.1.1: Transverse wave along a rope
A transverse wave travelling along a rope, travelling wave, has vibration direction perpendicular to the direction of propagation, e.g. wave on a rope, electromagnetic waves, light and radio waves. When this sine curve moves along its time axis with uniform velocity v, it represents a simple harmonic transverse progressive wave transmitting its energy with velocity v in the direction of motion. The arrangement of the particles along the line of advance at any instant is that of the sine curve. Every particle as the wave passes executes SHM in a direction at right angles to the direction of propagation, the amplitude is the same but there is progressive change in phase from particle to particle. The relationship between the velocity of propagation of the wave, its wavelength, and frequency is given by v = nX. Since n complete waves, each of length X, pass any point each second, i.e. the wave is propagated a distance nX each second.

| See diagram 25.1.1a: Transverse wave at time t and t + 1
Transverse waves travel so that the medium moves only at 90o to the direction of wave travel, move at a constant speed no matter their shape or size but individual parts of the medium move with varying speeds. In periodic waves the wavelength the distance between adjacent corresponding points, the frequency is the number of complete wavelengths that pass any given point per unit time, the period is the time interval for a complete wavelength to pass any given point, the amplitude is the maximum displacement from mean position and the speed is the distance travelled by the wave per unit time or frequency × wavelength. The frequency of a periodic wave is the same as the frequency of the generating source. A transverse wave travelling along a spring diminishes in amplitude as it dissipates energy. The speed for a transverse wave in a long spring =√ (tension in spring / mass per unit length of spring.) When waves approach each other, the sum of the individual displacements equals the resultant displacement at any time. When two wave shapes at 180o point rotations to each other move towards each other, the point at which they first meet, permanent rest point or node, remains stationary while the wave shapes move through each other.

Experiments
1. Attach coloured pieces of cloth to a rope at regular intervals. Tie one end of a rope to a support. Hold the other end so that the rope does not touch the ground. Make waves travel along the rope by moving the end of the rope up and down to make vertical waves, or moving left and right to make horizontal waves. Hold the rope still then strike the rope rhythmically with a stick to produce waves in the rope. Describe the motion of each coloured piece of cloth when a wave travels along a rope. Observe the difference between the motion of one coloured piece of cloth and the piece next to it when the wave travels along the rope.

2. Generate a wave along a rope by the sinusoidal vibration of the hand holding one end. The wave seen records the earlier vibrations of the source. Energy is carried by the wave from the source towards the right, along the rope in the direction of propagation (line of propagation) of the wave. Each particle of the rope vibrates perpendicular to the line of propagation so it is called a transverse wave. The speed of a transverse wave on a stretched string or wire is v = √ (tension in string / mass per unit length of string).

3. Waves are the propagation of vibration. Prepare a thin and stiff rope (similar to string) and tie some pieces of colour cloth to the rope every certain space. Fix one end of the rope, hold the other end of the rope by hand. Shake the rope up and down to form a vertical wave travelling along the rope. Shake the rope right and left to form a horizontal wave. Pull the rope by hand strength, beat the rope by a bar in a rhythm, also can produce a wave on the rope. Beat the rope by bar, first up and down, then right and left, the position of beating should be near your hand. Observe and describe how each piece of cloth moves as the wave travels along the rope and how different between two pieces of cloth next to each other. Shaking or knocking rope can produce the vibration at one end of the rope. The state and energy of this kind of vibration can be propagated from the vibrating source to the other end of the rope to form waves. During propagation of the waves, each point on the rope vibrates with the vibration of the source, if the source vibrates up and down each point on the rope also vibrates up and down, if the source vibrates right and left each point on the rope also vibrates in the same way. The amplitude of the vibration of each point is also confined by the source, if the amplitude of the source is bigger and the amplitude of each point is also bigger, the smaller the amplitude of the source the smaller the amplitude of each point is. Although each point on the rope only vibrates near its equilibrium point, but this kind of vibrating form and energy propagates as relay from point to another point next to each other and form waves of the whole rope. During propagation of the waves, the motion of points next to each other have either the same factors (amplitude, frequency) or difference. The point in just the position of the source vibrates first, which drives the point next to it, then the vibration is propagated to another point, gradually form waves. So the vibration of each point at the beginning has the difference of early or late, the times of arriving maximum amplitude are also different. In the other hand, in this experiment whatever the rope vibrates up and down or right and left, the vibrating directions of points on the rope are all vertical to the direction of propagation. Such a wave is called a transverse wave. The rope can only propagate transverse waves.

25.1.2 Reflected wave with a rope
Reflection is the throwing back or deflection of waves, such as light or sound waves, when they hit a surface. The law of reflection states that the angle of incidence (the angle between the ray and a perpendicular line drawn to the surface) is equal to the angle of reflection (the angle between the reflected ray and a perpendicular to the surface). Waves in a rope that reflect from a fixed end undergo a phase change of 180o, so they reflect upside down. Waves that reflect from a free end do not undergo a phase change of 180o, so they reflect same side up.
Experiment
Shake a rope and note how many waves form on the rope. Observe the wave propagation along the rope. The wave crests move towards the fixed point at other end of the rope then reflect. The original wave is called the incident wave. The frequency and amplitude of the reflection wave is the same with the incident wave.

25.1.3 Stationary wave, standing wave, with a rope or string
See: Standing waves, School of Physics, University of New South Wales. [Animation needs Flash 8 Plugin.]
See diagram 26.3.0: Standing waves in a string | See diagram 26.3.01: Standing waves in air columns
See diagram 26.3.2.1: Standing waves in a flute | See diagram 25.11: Stationary waves
Stationary waves occur under the following conditions:
1. two progressive wave trains are travelling in opposite directions with the same velocity
2. with equal wavelength
3. with equal amplitude.
One wave progresses to the right and the other progresses to the left.
 Stationary waves, shows one wave moving to the right and the other wave moving to the left. At times, 0, T / 8 and T / 4, the compounded effect is a stationary wave. The wave does not travel forward, but undergoes periodic changes of amplitude with the frequency of the original waves, the maximum amplitude being twice that of either wave. Each particle executes a SHM. the phase being the same for all the particles, but the amplitudes different, so that at any instant the curve show in a instantaneous displacement plotted against distance along the wave is a sine curve.
However, the curve does not travel forward. In a longitudinal stationary wave the displacements are along the direction of propagation of the wave and not perpendicular to it.
A node is a place where there is no motion.
An antinode is a place of maximum displacement. The distance between two adjacent nodes = α / 2. The distance between two adjacent antinodes = α /  2.
Experiments
1. Shake the rope to form four wave crests on the rope. Measure the length of the rope that vibrates, divide by four and attach red cloth at these points. Shake the rope to form four wave crests and observe the difference between the motions of the red cloth. The pieces of red cloth are almost static. The vibration of each point on the rope depends on both incident waves and reflection waves. The result of the piled up two waves makes that some points vibrate strongly and other points vibrate weakly or not at all, producing stationary waves.
2. Attach a 2 metres in length light string to a motor driven eccentric cam. The string passes from the electric motor then over a pulley with a 100 gm. weight attached to the end hanging down. Increase the speed of the electric motor to see standing waves with up to obtain 6 or 7 nodes.
3. Attach a light string the vibrating head of an electric toothbrush, then passed over a pulley and attach weights to vary the tension in the string. If the string is coated with fluorescent paint, the modes of vibration is easily seen in ultraviolet light.
3. A metre long pipe has small holes drilled equidistant along its length. One end of the pipe is connected to a natural gas supply. At  the other end is a horn loudspeaker with a signal generator and power amplifier. Adjust the gas supply to give a flame height of 5 cm. At different frequencies, standing waves form with height of the flames showing characteristic pressure maxima and minima in the pipe. The following frequencies produced the indicated modes (wavelengths): 200 HZ 1 / 4 wavelength, 400 HZ 1 / 2 wavelength, 800 HZ 1 wavelength, 1600 HZ 2 wavelengths.
25.1.4 Size of slit with a rope
See diagram 25.1.4: Rope through a slit
Waves of two different polarization can be exited on a rope. The rope can vibrate vertically or horizontally. If a rope passes through a slit slightly wider than the rope in a stand, only waves with vibration parallel to the slit can pass through it.
Experiment
Pass a rope through a vertical slit in a stand. Hold the end of the rope, shake up and down, and observe the wave propagation. Hold the end of the rope, shake it left and right, and observe the wave propagation. If the rope vibrates up and down, the rope does not touch the sides of the slit, which is the same case with no stand. If the rope vibrates right and left, as the vibration propagates to the slit the motion of the rope is confined by the slit and the vibration is resisted. The vibration energy transfers to the stand to make the wave energy decreases rapidly. Adjust the width of the slit and amplitude of the vibration to show how the amplitude of the waves gradually decreases. With the decreasing of the width of the slit gradually, the wave behind the slit disappears completely. You can also remain the width of the slit unchanged (but should be suitable), and change the amplitude of the rope to show the above phenomenon.

25.2.1 Transverse wave with a coiled spring, slinky spring, pulse
S05 Slinky Spring, wave modeller, "Prof Bunsen", (commercial website)
Wave demonstration springs, "Scientrific", (commercial website)
Experiment
See diagram 25.2.1
: Transverse wave with a coiled spring
The speed of the transverse wave in a spring =√ tension in the spring / mass per unit length of the spring. A pulse is a solitary wave disturbance generated by a single action.
Experiment
Hold one end of the long screw spring, let a student hold the other end of it and stand in a place about 10 metres far from you, ensure the spring is stretched between you. Try to shake your hand up and down until send a clear pulse along the direction of the spring. Several continuous pulses will consist of a series of transverse waves. Observe the direction of propagation of the pulses and vibration direction of the coils which are composed of the spring. Note that the amplitude of the transverse wave decreases as it travels along the spring due to loss of energy for stretching and contracting the metal in the spring.

25.2.2 Longitudinal wave with a coil spring, slinky spring
S05 Slinky Spring, wave modeller, "Prof Bunsen", (commercial website)
Wave demonstration springs, "Scientrific", (commercial website)
See diagram 25.2.2: Particles in a longitudinal wave
The diagram shows the successive instantaneous position of a row of particles, 1 to 15, at intervals of 1 / 8 of a period. The arrows at the left show the direction of movement of particle 1 between the instantaneous positions, particles in a longitudinal wave. Longitudinal waves travel so that the medium vibrates in the same direction as that in which the waves travel as a series of rarefaction and compressions.

See diagram 25.2.2a: Distance displacement graph for longitudinal SHM sound wave
The diagram shows a distance displacement graph for longitudinal SHM sound wave. Pressure variation in a longitudinal wave motion. The wave passes from left to right over a row of particles on the x-axis, distance axis. The graph shows the instant when the particle at the origin, 0, is not displaced. The displacement of a particle in a longitudinal wave is said to be positive when it is to the right of its mean position and negative when it is to the left of its mean position. The particles between 0 and 2. have all been displaced to the right, i.e. towards 2. The particles between 2. and 4. have been displaced to the left, i.e. towards 2. So the particles near 2. are closer together than usual. A compression is centred at 2. and the pressure near 2. is above normal. Similarly 6. to 11. is a region of increased pressure. Point 4. is a region of reduced pressure, a rarefaction. Particles between 2. and 4. are displaced to the left, i.e. away from 4. Particles between 4. and 6. are displaced to the right, i.e., also away from 4. Point 8. is at the centre of another zone of reduced pressure. At a point corresponding to 1., where the displacement curve is nearly flat and parallel to the x-axis, the displacements of adjacent particles are almost equal and, all to the right, i.e., the particles, although displaced, have their normal spacing, so 1. is a region of normal pressure. Points 3., 5., 7. and 10. are also regions of normal pressure. The direction of propagation of the wave, i.e. along the x-axis, the pressure is normal at 1., increases to a maximum at 2., decreases again to normal at 3., falls to minimum at 4., and is again normal at 5.
See diagram 25.2.2a: Pressure distance graph
In longitudinal wave motion the pressure is normal where the displacement is a maximum but has its maximum and minimum values where the particles are not displaced. The points of maximum and minimum pressure travel with the same velocity as the other attributes of the wave. Note the progression of the compressions and rarefactions.
Experiment
1. Restore the long spring to original state of stretching. Insert the other hand of yours into the spring coils, the position of inserting of your other hand is about 20 to 30 cm from the hand that holds the end of the spring. Press the coils between two hands to end of the spring, then remove the hand rapidly to make a longitudinal pulse propagate along the spring. Observe the direction of pulse propagation and the vibration direction of the coils.

25.2.3 Reflected wave in springs
Send a transverse wave as in experiment 25.2.1. Restrain the hand of the student who holds the other end of the spring stationary.
Experiment
Observe how the transverse wave be reflected from the end of the spring. At a junction of a heavy spring and a light spring an incident wave will be partially transmitted and partially reflected. When the incident wave is in the lighter spring, the reflected pulse undergoes phase change but the transmitted pulse does not. The reduced amplitude of the reflected pulse is caused by energy transferred to the heavier spring. When the incident wave is in the heavier spring there is no phase change for either the reflected wave or the transmitted wave.

25.2.4 Interference of waves in springs
Interference in one dimension, e.g. standing waves, path difference associated with the separation of two point sources, path difference = dxn / L, nodal and antinodal lines associated with two point source interference, constructive and destructive interference of two point sources. Interference occurs when waves from different sources interact with each other. Constructive interference, or reinforcement, occurs when a crest from one source meets a crest from another source, or when a trough from one source meets a trough from another source. Antinodes are points of constructive interference. Destructive interference occurs when a crest from one source meets a trough from another source. A node is a point of destructive interference. A nodal line is a line of consecutive nodes. If point sources of vibration are in phase, they produce an interference pattern that is "in phase". When two point sources, e.g. dippers in water, have the same frequency, but don't dip together, they produce an "out of phase" interference pattern.
Experiment
1. Two persons who hold two ends of the spring, shake the spring at the same time up and down, each send a transverse wave. Observe the phenomenon of across each other when and after two waves meet each other. This may need to exercise several times. After success, you can send a wave which is produced by shaking right and left at the same time, or one shake left, the other shake right at the same time, observe the case when and after two waves meet.

25.2.5 Wavelength, frequency and speed of wave in a spring
Send a series of waves by shaking the spring, the frequency of the hand is that of the wave. Send out a series of transverse waves. The distance between wave crest next to each other is equal to the wavelength.
Experiment
Observe the variations of the wavelength under different conditions of shaking slowly and rapidly (but ensure the series of waves are sent out). To measure the speed of the waves. Measure the actual length of the stretched spring S, exercise to observe the producing of the first crest in series of waves and trace it until it arrives to the end of the spring, then measure the time intervals t the first crest taken from this end to the other end of the spring. The speed of the wave is V = S / t.

25.2.6 Wave goes through from one medium to another medium in springs
Experiment
Use two screw springs in different diameters, connect them at one end, this can be regarded as two touching mediums. If you cannot find two different springs, you can use a rope replace one of the spring. Operate as in 25.2.1. Observe the behaviour of one transverse wave at the boundary of the two springs. Then send a series of waves which is stable in frequency (this need to exercise in advance, to maintain your hand shaking in a stable frequency), observe after wave goes into another medium, which quantities of wavelength, wave speed and frequency remain unchanged.

25.3.1.0 Ripple tank, wave tank
Waves, (ripple tank), "Scientrific", (commercial website)
See diagram 25.183: Ripple tank | See diagram 25.184: Ripple tank
See diagram 25.3.8: Single slit interference | See diagram 25.3.9: Two source interference (University of Melbourne)
Surface waves can be generated by disturbances on the surface of water of uniform depth. All surface waves travel at a constant speed in a uniform depth of water. A line disturbance in a uniform depth of water produces a straight wave which moves in a direction perpendicular, at 90o, to the wave front. In an uniform depth of water, all surface waves travel at a constant speed. A common misconception is that the size of sea waves increases in a regular fashion until the 9th or 10th wave.
Experiments
1. The ripple tank uses ripples on a water surface to demonstrate the properties of waves. It contains a shallow dish of water with a transparent base. An electrically operated vibrator dips into the water to generate the ripples. To simulate two source interference place objects in front of the source and use single or double prongs attached to the dipper to simulate two source interference. The commercial ripple tanks are available for use in schools include, e.g. Griffen, IEC and Pasco.

2. The ripple tank has with a glass bottom and it can hold water. Put the light source under the tank to see water ripples on the ceiling or put the light source over the tank to see water ripples on a sheet of paper below the tank. Use the tank in a dark place and where there is no vibration and no chance of anyone bumping into it. Adjust the depth to obtain only the required ripples. Fit sloping "beaches" of wire gauze around the edge. Note the circular pattern of ripples produced when a drop of water falls on the water in the tank. Use straight barriers and curved barriers with the height greater than the depth of the water and do not float.
3. A wave tank is composed of four parts: water tank, light source, vibration source, block boards (barriers). The size of the water tank is 45 cm in length, 30 cm in width, 10 cm in height. The bottom is made of glass. Put the water tank in a dark room. Allow no shaking outside the room or take measures to avoid shock. Pour five mm depth water in the tank. Let a drop of water fall on the surface of the water in the tank and observe the ring waves form on surface of water. If the reflection waves in walls of the tank are too strong, install some inclined metal net "beaches" around the walls of the tank to decrease the strength of the reflection waves. Use a moveable light source that can be placed on the bottom of the tank to shine the pattern of water waves on the ceiling, or placed in upper part of the tank to project the pattern on a piece of paper under the tank. Make a vibrator by attaching a piece of L-shaped thick wire to one end of a hacksaw blade. Clamp the hacksaw blade so that the end of the wire dips into the water. Pluck the end of the hacksaw blade and notice the circular waves formed in the water. For straight waves attach a T-shape piece of tin to the end of the hacksaw blade. Attach the L-shaped piece of wire or T-shape piece of tin to the armature of an electric bell. The barriers include straight and curved shapes with heights greater than the depth of the water and they do not float.
4. Make a vibrator
See 32.5.4.4: Electric bell
4.1 Attach a piece of L-shaped thick wire to one end of a hacksaw blade. Clamp the other end of the hacksaw blade so that the end of the wire dips into the water. Pluck the end of the hacksaw blade and notice the circular waves formed in the water. For straight waves, attach a T-shape piece of tin to the end of the hacksaw blade.
4.2. Attach an L-shaped piece of wire or T-shape piece of tin to the armature of an electric bell.
4.3 If you haven't a electrical vibrating apparatus, you can make one by yourself. Use a 30 cm long steel saw blade, fix a L-shaped thick wire at one end of the blade. Fix the saw blade to let the sharp end of the wire immerse into water. Pluck one end of the saw blade, observe that a ring-shaped waves formed on the surface of water. If you want to produce a straight line waves, use a T-shape small metal piece replace the wire. If you have a electrical bell, only fix L-shaped wire or T-shape metal piece on bell's armature an electrical vibrating source is made.

25.3.1.2 Straight barrier (straight block, plane barrier, block boards)
See diagram 25.183: Ripple tank
In an uniform depth of water in a ripple tank, a line disturbance produces a straight wave that moves on a direction at right angles to the wave front. When a straight wave reflects at a plane barrier, the angle of incidence = angle of reflection. A point disturbance in a uniform depth of water produces a circular wave which expands radially in all directions. When a circular wave reflects at a plane barrier each segment of the incident wave front obeys the law of reflection at the barrier. The reflected wave is circular and appears to be expanding radially from a point as far behind the barrier as the point disturbance was in front of the barrier. Surface waves refract towards the normal when crossing from deep water into shallow water. For a given pair of depths of water, sin i / sin r = speed in deep / speed in shallow = wavelength in deep / wavelength in shallow The wave frequency is the same in both deep and shallow water. The height of the block board should be higher than depth of water, should be various shaped boards of straight and curved. The board cannot be floating on surface of water. So the board should be made by heavy wood or plastic board.
Experiment
Prepare more same-shaped boards to be used in different experiments.

25.3.2 Simple straight line pulses, straight wave
See diagram 25.1.3: Straight wave in a ripple tank | See diagram 25.183: Ripple tank
1. Observe ripples hitting a straight barrier or the wall of the ripple tank, 1.1 as a circular pulse, 1.2 as a straight pulse hitting the wall at an angle of incidence smaller and greater than 45o.
2. Make pulses by giving a cylindrical wooden rod a sharp roll forward and back in the ripple tank. This motion produces continuous waves. The ripples are wider near the rod but sharper as they move away. The ripples are sharpest when the filament of the lamp is parallel to them.
3. Prepare a rectangle shape water tank with a flat bottom, pour a suitable depth of water in it. Roll a cylinder-shaped wooden bar back and forth on the surface of water, a straight line wave can be produced, repeat above action continuously, a continuous straight line of wave may be produced.
4. Use T-shape small metal piece, you can also produce a straight line wave in the tank.
5. Hang a long slice of wood by thin thread, pull the thread by hand and shake up and down to vibrate the wood on the surface of water, thus you can also produce a straight line wave in the tank. Near the vibrating source such as wooden bar, the wave area of the straight line wave is wider but as it goes forward it becomes narrow gradually. The main reason is that there is reflection waves produced on the walls of the tank to cause pile up of the incident waves and reflection waves forming a effect of "wave eliminating". If you change the amplitude and frequency of the vibrating source, you can change the shape of the straight line waves. When amplitude of the vibrating source is increased, the wave crest becomes higher, the trough becomes deeper. As the frequency becomes higher, the distance between the wave crest and trough decreases.

25.3.3 Simple circular pulses, ring-shaped wave
See diagram 25.183: Ripple tank | See diagram 25.3.9: Ripples crossing over
1. Touch the water with the following:
1.1 a finger,
1.2 a pencil point,
1.3 a drop of water from an eye dropper.
Note a single circular ripple in the middle of the tank. Make several such ripples one after the other.
2. Touch the water simultaneously in two places. Note the circular ripples crossing over each other.
3. Prepare a rectangular water tank with a flat bottom, pour water with a suitable depth in it. Use end of your finger or water drop comes from a pen or a dropper to produce a ring-shaped wave in centre of the tank.
4. As in the diagram, use L-shaped thick wire fixed on one end of the steel saw blade, you can also produce a straight line wave in the tank. Repeat the experiment several times, observe the variation of the amplitude of the wave crest. Change amplitude and frequency of the vibrating source, repeat again several times, observe the variation of the ring-shaped wave. The vibrating source of the ring-shaped wave on the surface of water is a point, while the wave propagates on a plane. With the distance from the source increases, the area of wave surface increases, the energy propagated from the source distributes on a larger area gradually, but the energy distributed every unit area decreases accordingly, so the amplitude of the wave crest decreases evidently which is just the need of conservation of energy. This is obvious different from the straight line wave on the surface of water. The vibrating source of the straight line wave is a line shaped thing, the wave surface is also a straight line, so the wave surface is not changed basically and the wave crest varies a little.

25.3.3.1 Reflection of circular wave at plane barrier
See diagram 25.3.3.1: Reflection of circular wave
Touch the surface of the water in a ripple tank with a pencil point to be the original point source of a circular wave. When the circular wave hits a barrier each segment of the incident wavefront obeys the law of reflection. The reflected wave is circular and appears to be expanding radially from a point behind the barrier. The distance from this point to the barrier is equal to the distance from the original point source to the barrier. It appears as if each reflected wavefront is being generated at an image source, S', which is as far behind the barrier as the original point source, S, is in front of the barrier. During the reflection of circular wavefronts off of a straight barrier, neither the frequency, wavelength, nor wave shape change. 

25.3.4 Reflection of waves at a straight barrier (straight block)
See diagram 25.2.1: Reflection of waves at a straight barrier | See diagram 25.3.4: Photograph | See diagram 25.183: Ripple tank
1. Observe ripples hitting a straight barrier or the wall of the ripple tank: 1.1. circular pulse, 1.2 straight pulse hitting the wall at right angles with straight pulse hitting the wall at an angle of incidence smaller and greater than 45o.
2. Waves can be reflected by a block, the reflected angle is equal to the incident angle. Prepare a water tank as in diagram 25.183. Use L-shaped wire vibrate continuously to produce a ring-shaped wave. Vibrate only one or two times to produce a ring-shaped pulse. Then observe the phenomenon happened as the wave and pulses strike the walls of the tank. Use T-shape board to produce a straight line wave and pulses. Observe the phenomenon happened as wave and pulses strike the walls of the tank. Put T-shape board parallel to the wall of the tank, vibrate once up and down to produce a straight line shaped pulse and make the pulse strike the wall of the tank vertically. Observe the phenomenon happened as the pulse strikes the wall of the tank. Make the T-shape board inclined about 45o to the wall of the tank, vibrate once up and down to produce a straight line shaped pulse and make the pulse strikes the wall of the tank in an angle slightly less or more than 45o. Observe the phenomenon happened as the pulse strikes the wall of the tank. Measure the angle between the incident wave surface and wall of the tank, that is the incident angle. Measure the angle between the reflected wave surface and wall of the tank, that is the reflected angle. Compare the magnitude of the incident angle and reflected angle. Put a straight line shaped block in the tank, remain the angle between T-shape board and wall of the tank, change the angle between the straight line shaped block and T-shape board, repeat the steps 4 to 8. Compare and verify for the results in step 8 and 9. In order to measure the angle between incident wave and reflected wave, you can put a straight bar on the tank to make the bar parallel to the wave surface of the incident wave, measure the angle between bar and wall of the tank or block, you can obtain the incident angle of water wave. Then put other straight bar parallel to the wave surface of reflected wave, measure the angle between the bar and wall of the tank or block, obtain the reflected angle. Then change the direction of T-shape board or block, repeat the experiment above. The angle that wave reflected on the surface of the block is equal to the incident angle. You can conclude that the wave can be reflected by a block and the reflected angle is equal to the incident angle.

25.3.5 Reflection of waves at a curved barrier, curved block
See 2.0.5: Conic sections, parabola | See 2.0.6: Parabola equation | See diagram 25.183: Ripple tank
1. Observe ripples hitting a circular barrier, 1.1 on the outside, 1.2 on the inside. Repeat the experiment with lens-shaped barriers.
2. Insert some thick copper wires in rubber tube, then bent the tube into a shape like a parabola to be a block board. As the thick copper wires increase the weight of the tube, you can fix it in water tank. Observe the reflected pulses produced by this kind of curved block.

25.3.6 Refraction of waves
See diagram 25.3.6: Photograph | See diagram 25.183: Ripple tank
Refraction is the bending of a wave of light, heat, or sound when it passes from one medium to another. Refraction occurs because waves travel at different velocities in the different media. Refractive index is a measure of the refraction of a wave as it passes from one transparent medium to another. If the angle of incidence is i and the angle of refraction is r, the refractive index n is given by n = sin i / sin r. It is also equal to the wave speed in the first medium divided by the wave speed in the second, i.e. n = v1 / v2. Refractive index varies with the wavelength.
Experiments
1. Put a plate of glass in the middle of the ripple tank to create a sloping depth. Observe the distance between crests, wavelength, as the depth becomes more shallow. The wavelength is less and the velocity of the wave is also lower in the shallow water than it is in the deep water.
2. As the speeds of the wave propagation in different medium are different, so in the boundary of the two mediums the direction of the wave propagation will change and produce the phenomenon of the wave refraction. Prepare a ripple tank as in the diagram. Put a glass board in the centre of the tank, the glass board lies on bottom of the tank, i.e. the board is parallel to bottom of the tank. Adjust the water level by a drinking straw to make the water being just immerse the glass board. Observe the variation of the wave surface as it goes through the boundary of the glass board. Compare the difference between the waves outside the glass and on the glass. Use another glass board that is different in side from that of the above, repeat the experiment above and observe. The glass board immersed in water forms a surface different from water. When the wave propagates to this boundary, the refraction will happen. If you observe, you will find that as wave goes through the glass board, the distance between the wave crest (i.e. wavelength) decreases. You can also see that the speed of water waves on the glass board (the place of water being shallow) is slower than original one, the place of water being deeper. Use the experiment to study the relationship between wave speed, wavelength and frequency. If you make the glass board into a certain shape, you can study the effect similar to triangular prism and lens.
25.3.7 Diffraction at narrow openings in barriers
See diagram 25.3.7:
Photograph | See diagram 25.183: Ripple tank
When straight waves pass through a narrow gap, or meet a small obstacle, the waves radiate through the gap, and curve behind the obstacle, called diffraction. Waves diffract more as the ratio wavelength / size of gap increases.
Experiments
1. Observe diffraction when a wave hits two barriers separated by a gap of 2 cm or less Repeat the experiment with two gaps. Repeat the experiment with more than two gaps equally separated. Repeat the experiments with different size gaps.
2. Each point on a wave surface is quite like a smaller wave source, its state of vibration and energy will spread in all directions to form a new wave surface. Prepare a ripple tank as in the diagram. Fix and install two straight line shaped block board in the tank about five cm from the vibrating source, and leave a slit less than two cm between them. When water wave propagates to the block, you can observe the diffraction wave behind the silt. Change the width of the slit, observe again the diffraction wave of the water. If the frequency of a water wave is higher, you can only use a stroboscopic equipment to observe. During high frequency, the block itself may produce vibration to disturb the diffraction of a water wave. The wave from the side of the block board will destroy the diffraction. Observe by changing the width of the slit that the wider the slit, the more unclear the diffraction phenomenon.

25.10 Speed of wave propagation
Speed of wave propagation, c, is the distance travelled by a wave in unit time. Distance = velocity × time, so wavelength = v × T = v / f. Velocity, v, = frequency × wavelength (f × λ).
1. Velocity of longitudinal progressive waves in an elastic medium, v = √ E / ρ, where E is the appropriate elasticity, e.g. the Young's modulus for a rod, or the adiabatic bulk modulus (yP) for a gas. and ρ = density of the material.
2. Velocity of transverse waves along a stretched string, v = √ F / µ, where F = tension in the string, µ = mass of the string per unit length.
3. Velocity of transverse waves in a vacuum, v = 3 × 1010 cm. / sec. The energy of a wave is proportional to the product of the square of the amplitude and the square of the frequency.

25.11 Stationary waves, standing waves
See: Standing waves, School of Physics, University of New South Wales. [Animation needs Flash 8 Plugin.]
See diagram 26.3.0: Standing waves in a string | See diagram 26.3.01: Standing waves in air columns
See diagram 26.3.2.1: Standing waves in a flute
See diagram 25.11: Stationary waves
Stationary waves occur under the following conditions:
1. two progressive wave trains are travelling in opposite directions with the same velocity
2. with equal wavelength
3. with equal amplitude.
One wave progresses to the right and the other progresses to the left.
 Stationary waves, shows one wave moving to the right and the other wave moving to the left. At times, 0, T / 8 and T / 4, the compounded effect is a stationary wave. The wave does not travel forward, but undergoes periodic changes of amplitude with the frequency of the original waves, the maximum amplitude being twice that of either wave. Each particle executes a SHM. the phase being the same for all the particles, but the amplitudes different, so that at any instant the curve show in a instantaneous displacement plotted against distance along the wave is a sine curve.
However, the curve does not travel forward. In a longitudinal stationary wave the displacements are along the direction of propagation of the wave and not perpendicular to it.
A node is a place where there is no motion.
An antinode is a place of maximum displacement. The distance between two adjacent nodes = α / 2. The distance between two adjacent antinodes = α /  2.

25.12 Annulment of waves
See diagram 25.11: Annulment of waves, shows that the resultant of the two waves at all places at all times is zero.
Annulment of waves, interference, occurs when: 1. two wave trains travelling in the same direction, 2. with equal wavelengths, 3. equal velocities, 4. equal amplitudes, and, 5. phase difference of half a period, i.e., half a wavelength.

25.13 Beats
See diagram 25.11: Beats, shows that the curve, in which displacement is plotted against time, obtained by compounding the two waves has a varying amplitude, it rises to a maximum, falls to a minimum, and rises to a maximum again in the time required for A to gain one complete vibration, i.e. in 1 / 12th second. Thus there is one " beat " in 1 / 12th second, and in one second there are 12 beats. The number of beats per second equals the difference in the frequencies of the notes.
Beats occur when 1. Two wave trains travelling in the same direction with the same velocity, 2. wavelengths are nearly equal, 3. amplitudes are equal, for maximum effect.
The energy rises to a maximum and falls to a minimum periodically. These pulsations in energy are called beats. If two wave trains A and B, have frequencies 60 per second and 48 per second, in 1 / 12th of a second A makes 5 complete vibrations and B makes 4 complete vibrations.
25.14 Doppler effect
See: Doppler effect, School of Physics, University of New South Wales. [Animation needs Flash 8 Plugin.]
The Doppler effect is caused by the change in pitch due to relative motion between sound source and listener. A listener approaching a stationary sources of sound will hear a higher pitched sound. A listener moving away from a stationary sources of sound will hear a lower pitched sound. You can hear the effect when a train rushes past while sounding its horn or whistle, or a car rushes past with the horn sounding or a police car rushes past with siren sounding. If a moving source of sound emits a sound of frequency fs, v = speed of sound, and the source of sound approaches the observer at speed vs, measured relative to the medium conducting the sound, an observer moving towards the source at speed vo will hear a sound of frequency fo = fs(v + vo/ v - vs). When the source and the observer approach each other, more wave crests strike the observer per second than when the source and the observer are at rest, causing the ear to perceive a higher frequency than the frequency emitted by the source. When the source and the observer move away from each other, less wave crests strike the observer per second than when the source and the observer are at rest causing, the ear to perceive a lower frequency than the frequency emitted by the source.
1. Doppler's principle applies to the effect of motion of source, observer and medium on the apparent frequency of a vibration. For simple harmonic progressive waves, let V = velocity of the disturbance in the medium at rest, i.e. the velocity of transmission, f = the frequency, λ = wavelength, T = the period So V = f × λ, f = 1 / T, λ = VT = V / f. Assume that all movement is positive and from left to right.
1. Source medium and observer are at rest
S at rest, medium at rest, wave ---> V (f = V / λ). Observer at rest.
Frequency f = v / λ

2. Source is moving, medium and observer are at rest
[If source only moving, λ unchanged, V unchanged]
S ---> u, wave ---> V, Observer at rest
The moving source does not affect the velocity of transmission of the wave and does not affect its apparent velocity to the observer, so V is unchanged. The wavelength and frequency are changed because the source is moving towards the observer. In time T source moves forward distance uT, so wavelength decreased by uT.
Changed wavelength, λ 1 = (λ - uT) = (V / f - u / f) = (V- u / f).
Let observed frequency = f'1, then V / f1 = (V- u / f).
Changed frequency, f1 = (fV / V- u).
The pitch of a note will be raised if the source be moving towards the observer and lowered if moving away, e.g., change in pitch of a train whistle on passing an observer.

3. Source is at rest, medium is at rest, observer is moving
[If observer only moving, V apparently changed, λ unchanged.]
S at rest, wave ---> V, Observer ---> v
Movement of observer does not affect velocity of transmission V. However, observer is moving away from approaching wave with velocity v, so the apparent velocity of the wave to the observer, V', is decreased. V' = V - v.
The actual wavelength, λ, is unchanged and is the same as the apparent wavelength. V1 = (V - v). wavelength λ is unchanged.
Let apparent frequency = f2.
Apparent frequency, f2 = (V - v / λ) = [n (V - v) / V], because λ = (V / f) ).

4. Source is moving, medium is at rest, observer is moving
[If source and observer moving, λ unchanged, V apparently changed.]
S ---> u ---> V, Observer ---> v
Movement of source and movement of of observer do not affect the velocity of transmission of the wave V. However, as observer is moving with velocity v to apparent velocity V1 to the observer is decreased. V1 = (V - v). Also, the wavelength λ is decreased to λ1 as in (b). Let new apparent frequency = f3. Apparent frequency f3 = (V1 / λ1) = [f(V - v) / V - u]
(e) Source, medium and observer are all moving
S ---> u ---> V, Observer ---> v
If the medium is moving with velocity w, apparent frequency, f4 = [f (V + w - v) / (V + w - u)]

5 .If the directions of any of the first three velocities are not parallel to that of V, then their components in the direction of V must replace the quantities themselves. The motion of the stars causes the frequencies of the lines in their spectra to be altered, and from the change in wavelength the velocity of the star relative to the earth can be determined and whether the earth and star are approaching one another or receding from one another.
Experiment
See diagram 26.2.4: Swinging hand-held speaker
1. Show how the frequency of a moving sound source changes because of the relative velocity of the source to the observer. For a sound source, use a battery-operated oscillator driving a loudspeaker at constant frequency f. Attach the sound source to a strong string. Hang on to the end of the string then swing the sound source around and above head. The pitch of sound from the sound source is heard to raise and lower as the the sound source moves towards or away from the observer. If the length of the string = 60 cm, rotational frequency = 4 revolutions / second, linear speed = 15 meters / second, so maximum doppler frequency 1.04 f = and minimum doppler frequency = 0.96 f.