Figure 6.1 Transverse and longitudinal waves on a slinky.

Figure 6.1 Transverse and longitudinal waves on a slinky.

Water waves can be a mixture of transverse and longitudinal vibrations. Individual elements of water move in circles or ellipses, oscillating both perpendicular and parallel to the surface of the water.

As sea waves approach a beach the motion of the particles changes as the depth of the water decreases. At a certain point the friction of the sand causes the wave to 'break'.

Sea storm in Pacifica. Courtesy of Mila Zinkova.
The waves appear to be coming in but they do not bring in the sea!

6.2 The mathematics of a travelling wave

6.2.1 The making of a wave

We base the mathematical representation of a wave on the assumption that travelling waves are made up of basic components oscillating continuously about given positions in space.

Let us choose a simple transverse wave in a material medium. We will call the maximum displacement of each oscillating particle the amplitude A.

6.2.2 From the sine of an angle to the picture of a wave

The sine of an angle is defined in terms of the ratio of two sides of a right angled triangle. Many physical phenomena, such as refraction, are described in terms of the sines of angles (e.g. Snell's law in Chapter 3). The function which expresses the value of sin(6) in terms of 6 is called the sine function. It is the basic function which represents the physical properties of periodic waves.

Generating the sine function

Let us increase the angle 6 in Figure 6.2a by rotating the hypoteneuse of the triangle ABC anti-clockwise. As 6 increases

from zero to 90° the value of sin(0) rises from 0 to 1, falls back to zero at d = 180° and then becomes negative as the side BC points in the negative y-direction. Finally, when d reaches 360° the function sin(0) returns to its initial value at d= 0. As we continue to rotate the arm AC in an imaginary circle, the function sin(0) continues to repeat in a periodic cycle as illustrated in Figure 6.2b.

If we rotate the radius at a constant angular speed o (radians per second), d changes at constant rate, and we can write d = ot. The graph then shows how the lateral displacement of a particle in a given place in the medium varies with time. Such a particle oscillates with simple harmonic motion represented by the function y = sin(ot).

In a continuous medium which can transmit transverse waves, any one particle will not oscillate in isolation but will transmit its motion to neighbouring particles resulting in the creation of a simple harmonic wave. The upper part of Figure 6.3 depicts a 'photograph' of such a wave giving the transverse displacements from equilibrium (y) of successive particles at a given moment in time. The profile of the wave is the function y = sin(x), where x is the distance in the direction of propagation (x). Note y profile of a sinewave y profile of a sinewave

the wave has moved on the wave has moved on y=sin(x-a)

Figure 6.3 Advancing sine wave.

Figure 6.3 Advancing sine wave.

that the pattern repeats after a distance A = 2n, the wavelength of the wave.

The lower curve is another snapshot, taken when the wave has advanced a distance a in the x direction. We represent this wave by introducing a phase difference = -a into the original expression. The first null point, which had previously been at the origin is now at the position x = + a. All other points on the wave have advanced by the same distance.

6.2.3 An expression for a sine wave in motion

So far our mathematical expression represents a wave frozen in time. It is a 'snapshot' which shows the wave at a given instant. We can represent a moving wave by letting the phase shift change with time at the rate a = vt where v is the speed of the wave.

Making adjustments to the scale

We adjust the equation to give the correct wave amplitude by multiplying the sine function by the factor A. We can also adjust

the scale so that the distance between wave crests is X instead of 2n by writing it in the form:

The value of this function is +A for x = 0, X, 2X, 3X, ..., etc. Combining the above with a phase shift which changes with time we obtain:

This represents a simple harmonic wave of amplitude A and wavelength X travelling with speed v from left to right along the x-axis.

6.2.4 Wave parameters

The time for any particle to complete one oscillation is called the period T while the number of oscillations per second is called the frequency f = 1T.

The wave travels a distance X in a time T so the speed of the wave v = — = Xf. T '

The wave equation 6.1 is often written:

where k = 2nIX is called the wave number and o = 2nvIX = 2nf is known as the angular frequency. (It is also the angular speed of the radius vector in the representative circle in Figure 6.2a).

wavefronts v

It is often convenient to picture a wave in terms of wave fronts, which are the surfaces joining points equally displaced by the wave at any one time.

6.3 The superposition of waves

6.3.1 The superposition principle

The superposition principle states that the total displacement of any particle, simultaneously disturbed by more than one wave, is simply the linear sum of the displacements due to the individual waves.

When droplets of rain fall on the surface of a pool, they create circular surface waves which expand and overlap one another. Each wave is unaffected by the presence of the others, and each independently displaces particles of water. To get the total displacement of any wavefronts

Raindrops. Courtesy of Piotr Pieranski.

particle, we simply add the displacements due to individual waves.

6.4 Applying the superposition principle

6.4.1 The superposition of two waves travelling in the same direction

Two identical sine waves travel in the same direction:

Waves in phase: the individual waves combine to give a wave with a total amplitude twice that of either wave.

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