Conceptual Physics

radians.

The equation in Equation 1 describes a transverse wave moving from left to right. For a

wave moving from right to left, the minus sign inside the phase is switched to a plus

sign, reversing the sign of the coefficient of time.

Equation 1 assumes that a particle at position x = 0 at time t = 0 is at the equilibrium

position y = 0. You can add what is called a phase constant to the equation to create a

new equation describing a wave with a different initial state. For example, suppose a

constant angle such as ʌ/2 radians were added to the argument of the sine function.

Then, the particle at x = 0 at time t = 0 would be at its maximum positive displacement,

because the sine of ʌ/2 equals one. A phase constant does not change the shape of a

wave, but rather shifts it back or forward along the horizontal axis by the same amount

at all times. Note that as the phase is increased by an integer multiple of 2 ʌ radians, the

sine function describing the wave behaves as if there were no change at all.

If you contrast the equation here to the equation for simple harmonic motion, you will

note that the equation for a traveling wave requires two inputs to determine the vertical

displacement of a particle. Both equations include time, but the equation in this section

also requires knowing the x position of a particle in the medium. With a wave, the

vertical displacement is a function not only of time, but also of position in the medium,

while the position is not a factor in SHM.

The equations to the right can be used with either transverse or longitudinal waves.

When applied to longitudinal waves, the oscillation of the particles occurs parallel to the

direction of travel of the wave, and then we would use the variable s instead of y to

represent the horizontal displacement of a particle away from its equilibrium position.

Equation for traveling wave

Function relates particle’s vertical

displacement y to:

·particle’s horizontal position x

·elapsed time t

·wave’s amplitude, wavelength,

frequency

y = A sin (2ʌx/Ȝí 2 ʌft)

y = particle’s vertical displacement

A = amplitude of wave

x = particle’s horizontal position

Ȝ = wavelength, f = frequency

t = elapsed time

For wave motion toward íx:

y = A sin (2ʌx/Ȝ + 2ʌft)

15.11 - Gotchas

A mechanical wave can travel with or without a medium. No. Mechanical waves must have a medium. This is why, in the vacuum of space,

there is total silence. Sound, a mechanical wave, cannot travel without a medium such as air.

The medium carrying a wave does not move along with the wave. That is correct. The medium oscillates, but it does not travel with the wave.

This differentiates wind, which consists of moving air, from a sound wave. With a sound wave, the air remains in place after the sound wave

has passed through.

The amplitude of a wave has no effect on the speed of the wave. That is correct. The speed of a wave is determined by the properties of the

medium. This means that if you are in a hurry, it is no use yelling at people!

Wave speed in a string is a function of frequency, so if I increase the wave frequency, the wave speed will increase, too. No. The speed of a

wave in a string is fixed by the tension and linear density of the string. Increasing wave frequency will cause a decrease in wavelength, but no

change in wave speed.

Amplitude is the same as the vertical displacement y of a particle in a wave. No, the amplitude Ais the maximum positive vertical displacement

of a particle, while at a time t the instantaneous vertical displacement y can be anywhere between +A and íA.

(^302) Copyright 2007 Kinetic Books Co. Chapter 15