Class content > Oscillations and Waves > Waves in 1D > Waves on an elastic string

Prerequisites:

• Complex dimensions and dimensional analysis
• Mass on a spring
• Waves on an elastic string
• Propagating a wave pulse - the math

We can wiggle an elastic string by moving our hand up and down in almost any way we want to propagate a signal of any arbitrary shape (as long as we don't move it so dramatically that we bend the string so much the small angle approximation no longer works). We've looked only at simple pulses so far -- basically a single moving bump. But a sinusoidal oscillation turns out to be a particularly useful one. If we oscillate our hand up and down like a simple (undamped) harmonic oscillator,

The position of the hand has been taken as x = 0. The result will be that a sine (or cosine)* wave begins to move out along the string, making the shape of the string at any instant of time into something that looks like a sine wave.

* For some reason the tendency is to use cosine when we are doing the simple harmonic oscillator like the mass on a spring and the sine when we are describing a propagating wave on a string. Maybe this is because we often start the SHO by pulling it back and releasing it so it often starts at its maximum like a cosine, and we look at strings that are tied down and have the value 0 on the end like a sine. It doesn't really matter and we can add in a phase shift to either example if we like .

We'll have more to say about this rather complicated structure at the end of this article. At a fixed instant of time, the result looks something like the figure below. (The figure below is clipped from the PhET program, Waves on a String. We highly recommend playing with this for a while.)

Why should we bother to look at periodic oscillations?

This seems on the surface to be a rather strange choice. Why should we be interested in looking at waves that look like sines or cosines? There are (at least) two rather important reasons.

1. Natural drivers of waves are often periodic oscillations -- Often, waves are started by something vibrating. Since the vibration of anything near to equilibrium often looks like a harmonic oscillator, the sources of waves often move like a harmonic oscillator. If the matter is vibrating in a medium (air or water) it will create sound waves. If the vibrating matter are charges (electrons in a molecule) it will create electromagnetic waves (light, radio waves, x-rays, etc.).
2. Essentially any wave shape can be expressed as a sum of sinusoidal oscillations -- This result is called Fourier's theorem and the expression of a signal as a sum of different frequencies is called a Fourier transform or a spectral analysis. This is an extremely powerful tool for analyzing any kind of propagating signal.

An example of the decomposition of a complex signal into frequencies is shown at the right. This is taken from a study of the signature whistles of bottlenose dolphins (Tursiops truncatus). Each adult dolphin has an individual characteristic complex whistle that it makes when it meets other dolphins or is in a stressful situation. The figure at the right was part of a study of whether a dolphin calf's signature whistle is inherited or learned, and if learned, from whom.** The whistle is short -- less than two seconds. To analyze the structure of the whistle in order to match is with individual dolphins, the researchers broke the signal up into 150 time bins (shown on the horizontal axis). Each bin contained many oscillations. The pattern was expressed as the sum of many different frequencies and their intensity is plotted on a vertical line above the time bin. Thus, dark spots show a contribution of that frequency to the whistle in that time bin. The upsweep of the dark line indicates a whistle with a rising pitch. ** D. Fripp & P. Tyack, "Postpartum whistle production in bottlenose dolphins," Marine Mammal Science, 24(3), 479-502 (2008).

Other applications of sinusoidal wave analysis includes NMR and molecular fingerprinting.

The math of the sinusoidal wave

Although the sinusoidal wave is generated by an oscillation in time at a fixed point in space {y(0,t)}, it's somewhat easier to make sense of the math if we think about what our elastic string looks like in space at a fixed time {y(x,0)}. We can see from the simulation that a sinusoidal driver at a particular point generates a shape that looks like a sine curve. We can figure out how we have to write this mathematically by doing dimensional considerations like we did for the SHO. (See Mass on a spring.) 

Getting the dimensions right

If we know the shape of our string at a fixed time (say t = 0) is like a sine then we might start by writing

y(x,0) = sin x

But we know we can't get away with this for dimensional reasons. We can't take the sine of a dimensioned quantity -- only of a ratio. Otherwise, changing (here) our length scale would yield a different result, and a physical result cannot depend on what arbitrary choices we make to measure with. Similarly, y is a distance but sin is defined as a ratio so it is dimensionless. We have to introduce constants of scale with appropriate dimensions. We'll first make the math work and then figure out what they mean.

If we define two constants, "A" having dimensions of length (L) and "k" having dimensions of inverse length (1/L) then we could write

y(x,0) = A sin kx

and have the dimensions all come out right.

Making it move

From our analysis of the motion of signals along a string (see Propagating a wave pulse - the math) we know how to make a stationary mathematical function move: we replace x everywhere in the argument by x - v₀t. This will make the function move in the positive x direction with a speed v₀. The result is

y(x,t) = A sin k(x-v₀t)

This form of a right-traveling sinusoidal wave is convenient for seeing that it is a wave moving in the +x direction. If we had instead used x+v₀t we would have gotten a left-traveling sinusoidal wave. {Note: We should really be more careful here. The "left and right" depend on our having chosen the positive x direction to be to the right. That isn't always the case.}

In order to separately see the space and time dependence for when the other variable is held fixed (we are looking at the space dependence at a particular instant of time -- a photo, or at the time motion of a particular bit of the string) it is more convenient to multiply the parentheses out.  We write

k(x-v₀t) = kx-kv₀t = kx-ωt

where for convenience we have defined the combination

kv₀  =   ω

This has dimension of (1/L)(L/T) = 1/T so it is just like the angular frequency we defined in our discussion of the SHO. This gives the result shown at the top of the article for a moving sinusoidal wave:

Many different forms of this expression are convenient depending on what we want to look at. (See the problem Equations for sinusoidal waves.)

Making sense of the equation

The equation for the displacement of an elastic string undergoing sinusoidal oscillation is   y(x,t) = A sin(kx-ωt)   What does it all mean? Let's make sense of each part of it.

1. What are we talking about?  The expression y(x,t) means that we are finding the y displacement of the bit of string that is labeled by its position x at a time t.

2. What's A? Since we know that the function "sin" only oscillates between 1 and -1 and is dimensionless, multiplying it by A means that y will oscillate between the values A and -A. So we can interpret A as the amplitude of the oscillation.

3. What's k? The constant k was just introduced to make units come out right. But it will have implications. If we fix t (say for convenience at t = 0) then our function is A sin kx. How does this change? We know that the sine goes through a full oscillation when its argument (in radians) changes by 2π (say, from 0 up to 2π). If kx changes by 2π, then x must change by 2π/k. Therefore, when x changes by 2π/k it goes through one full oscillation. The spatial distance for one full oscillation is called the wavelength, λ. Therefore

k = 2π/λ.

Choosing k selects how fast the wave will be changing in space.

4. What's ω? Similarly, when we consider a fixed x position (say for convenience x = 0) and look at the time variation of a particular bead, we get something proportional to sin ωt. We know that the sine goes through a full oscillation when its argument (in radians) changes by 2π (say, from 0 up to 2π). If ωt changes by 2π, then t must change by 2π/ω. Therefore, when t changes by 2π/ω it goes through one full oscillation. The time for a bit of the string to go through one full oscillation is called the period, T. Therefore

ω = 2π/T.

The inverse of the period is also a convenient variable, the frequency, f. It's measured in inverse seconds (cycles per second) which is called Hertz. This gives the equations

f = 1/T = ω/2π.

It's easy to see that this is right via unit conversion. The frequency f is in cycles/sec, the angular velocity  ω is in radians/sec and 1 cycle = 2π radians. So multiplying omega by 1 = (1 cycle)/(2π radians) converts the units from radians/sec to cycles/sec.

Relating the frequency and the wavelength

We've related the frequency and the wavelength to our variables ω and k, but we have a relationship between them: ω = kv₀. What does that tell us about the frequency, wavelength, and period? If we express ω and k in terms of frequency and wavelength in this relation, we get

ω = kv₀

(2πf ) = (2π/λ)v₀

fλ = v_0.

So the product of the frequency and the wavelength is the wave speed. This makes more sense if we express it in terms of the period. Since f = 1/T, we get

λ = v₀T.

This makes good sense. If we wiggle our hand that is generate the wave, in the time we go through one full oscillation (the period, T) a full wiggle will have run out onto the string -- one wavelength, λ.

Joe Redish 3/31/12