Sinusoidal waves


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,

$$y(0,t) = A \sin{\omega t}$$

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 .

$$y(x,t) = A \sin{(kx - \omega t)} = A \sin{k(x - v_0 t)}$$


$$v_0 = \frac{\omega}{k}$$

We'll see below why it has 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.
  3. Our body's physical sensors respond to oscillations with a well defined wavelength and frequency — We personally detect two kinds of waves in our environment to help us build a picture of the world we live in: sound and light. Our detectors for these waves — ears and eyes — are evolved to respond differently to different frequencies, giving us perception of pitch and color. As a result, an analysis in signals in terms of sinusoidal oscillations tends to make sense to us.
Source: D. Fripp & P. Tyack, "Postpartum whistle
production in bottlenose dolphins," Marine Mammal
, 24(3), 479-502 (2008)

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.

Other applications of sinusoidal wave analysis includes NMR and "molecular fingerprinting" (identifying a molecule by the spectral analysis of its emission and absorption of light).

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 dimensionless 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. The constant $k$ is referred to as the wave number of the oscillation.

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 $x - v_0t$. This will make the function move in the positive $x$ direction with a speed $v_0$. The result is

$$y(x,t) = A \sin{k(x-v_0t)}$$

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 $x+v_0t$ 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

$$k(x-v_0t) = kx-kv_0t = kx-ωt$$

where for convenience we have defined the combination

$$kv_0  =   ω$$

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 simple harmonic oscillator (mass on a spring). This gives the result shown at the top of the article for a moving sinusoidal wave:

$y(x,t) = A \sin{(kx - \omega t)} = A \sin{k(x - v_0 t)}\quad$ with $\quad v_0 = \frac{\omega}{k}$

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

Joe Redish 3/31/12



Article 690
Last Modified: June 10, 2019