Describing a Travelling Wave

Transcript

0:00Introduction

Hello everyone! Let's start with a small puzzle. At a cricket match, the crowd does a Mexican wave, and it races right round the stadium. But each fan only stands up and sits down, in their own seat, and nobody runs round. So what is it that travels? Keep that puzzle in mind. Reading a wave from its equation comes up often in JEE, so let's begin.

0:25A wave carries energy, not matter

Here is the answer to our puzzle. What went round was the disturbance, the standing up and sitting down, not the fans. That is exactly what a wave does: it carries a disturbance, meaning energy and momentum, but not matter. Okay? Now, there are two kinds of waves. In a transverse wave, the particles move perpendicular to the direction of travel, like a wave on a string, or our fans. So tell me, sound in air, which kind is it? It is longitudinal, yes, exactly! The air particles move back and forth along the direction of travel. Right?

1:02Any shape moving without change

Now, the equation of a wave. Here y is the displacement at position x and time t. If any shape slides towards plus x at speed v without changing, then y equals f of, x minus v times t, where f gives the shape. The minus sign is there because in time t, the shape moves forward by v times t. With me? Now the same shape slides the other way, towards minus x. What changes in the equation? Only the sign, that is correct! Towards minus x, y equals f of, x plus v times t. Got it? And every such wave satisfies this equation. The curly d means a partial derivative, keeping the other variable fixed, and the small two means the second derivative. So, the second partial derivative of y with respect to x equals one over v squared, times the second partial derivative of y with respect to t. Clear?

2:01The sinusoidal wave

Now the most important shape, the sine wave in the figure. y equals capital A times sine of, k x minus omega t plus phi. Capital A is the amplitude, marked on the right. Omega, the Greek letter like a rounded w, is the angular frequency, and phi, the circle with a line through it, is a constant phase. The short vertical red arrows come later. Okay? Next, k equals two pi over lambda. Lambda, the Greek letter at the top of the figure, is the wavelength, from one crest to the next. And omega equals two pi f, where f is the frequency. Right? Now take k common from k x minus omega t, and compare with f of, x minus v times t. So what is the wave speed v? k x minus omega t equals k times, x minus omega over k times t, in big brackets. So v equals omega over k, yes, exactly! And putting in omega and k, that is also f times lambda. Clear? And the direction is really simple. Look only at the signs in front of the term with x and the term with t. Opposite signs mean towards plus x, like the black arrow in the figure, and same signs mean towards minus x. Got it?

3:20Phase difference

Next, phase difference. The little triangle is capital delta, meaning a difference. For two points delta x apart, the phase difference, delta phi, equals two pi over lambda, times delta x. With me? And at one point, two moments delta t apart differ in phase by two pi over capital T, times delta t, where capital T is the period. Okay?

3:45Particle velocity is not the wave speed

Now a big idea. The wave moves along x, but each particle of the string only moves up and down. For a wave towards plus x, the particle velocity, the partial derivative of y with respect to t, equals minus v times the partial derivative of y with respect to x, which is the slope of the string. Right? So, take a point on a rising part of the curve, where the slope is positive. Use the formula: is the particle velocity at that point positive, or negative? It is negative, that is correct, because of the minus sign! So a point on a rising slope moves down, and a point on a falling slope moves up. Now look at the short vertical red arrows in the figure: they show exactly this. Clear? Now here is a classic trap. Is the largest speed of a particle equal to the wave speed v? No, it is not, and many students slip here. The largest particle speed is capital A times omega, which is not the wave speed v. Remember the stadium: the wave races round, but each fan just stands up and sits down. Got it?

4:54Example 1: A, f, lambda, v and direction of a sine wave

Now our first example. y equals zero point zero five times sine of, twenty pi t minus four pi x, in S I units, so metres and seconds. Find the amplitude, frequency, wavelength, speed and direction of travel. Pause the video and try it first. Done? Okay, let's check. Compare with the sine wave we just learnt. Capital A, the amplitude, is zero point zero five metres. Omega is the number with t, twenty pi, and k is the number with x, four pi. Right? So f is omega over two pi, which gives ten hertz. And lambda is two pi over k, which gives zero point five metres. With me? The speed v is omega over k: twenty pi over four pi gives five metres per second, and the small minus one on the s means per second. As a check, f times lambda is ten times zero point five, which also gives five. Okay? Now, here the term with t is written first. Does that change the direction? Not at all, because only the signs matter. Plus twenty pi t and minus four pi x have opposite signs, so the wave travels towards plus x. Writing t first is just a way to confuse you. Got it?

6:18Example 2: sin kx cos omega t vs sin squared (kx - omega t)

Now a harder one. Show that y equals capital A sine k x, times cos omega t, is not a travelling wave, but y equals capital A sine squared of, k x minus omega t, is one; the small two means the sine is squared. Pause the video and try it first. Done? Okay, the first one. The product to sum formula gives capital A over two, times, in square brackets, sine of, k x minus omega t, plus sine of, k x plus omega t. Using our sign rule, which way does each part travel? The first part goes towards plus x, and the second towards minus x. Two waves moving opposite ways cannot be one shape, f of, x plus or minus v times t, which is the plus or minus sign on screen. So it is not a travelling wave. Okay? Now the second one. The identity for sine squared turns it into capital A over two, times, in square brackets, one minus cos of, two times, k x minus omega t. Clear? Taking k common again, y depends only on x minus omega over k times t, which is the form f of, x minus v times t. So it is a travelling wave, moving towards plus x at speed omega over k. With me?

7:44Quick revision: f(x - vt), v = f lambda, phase, A omega

So, quick revision. First, a wave carries energy, not matter; f of, x minus v times t, goes towards plus x, and f of, x plus v times t, towards minus x. Second, k is two pi over lambda, omega is two pi f, and v is omega over k, or f times lambda. Third, the phase difference is two pi over lambda times delta x, and the largest particle speed is capital A times omega, not the wave speed. Now try the practice problems on this page. All the best!