# Reading the content in a sinusoidal wave

#### Prerequisites

## Making sense of the equation

The equation for the displacement of an elastic string undergoing sinusoidal oscillation is

$$y(x,t) = A \sin{(kx-ωt)}$$

This is an anchor equation. It can serve as the basis for understanding a lot of interesting phenomena, including beats, interference, and spectral analysis.

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 $t$. By labeling the displacement as $y$ and the position along the string $x$ we imply that the wave is a transverse wave. More generally we could write: $f(x,t) = A \sin{(kx-ωt)}$ where $f$ indicates the displacement of the bit of string away from its equilibrium position. The tricky part is that it oscillates in both time and space.

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 introduced to make units come out right. But it will have implications. Since $k$ is about how fast the argument of sine changes with $x$, let's choose a fixed value of $t$, say for convenience at $t = 0$. Then our function is $A \sin{kx}$. It looks like the figure below.

We've put a little ruler to indicate that the axis is a position measurement.

How does this change as $x$ changes? 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 $2π/k$ our function goes through one full oscillation. The spatial distance for one full oscillation is called the *wavelength**, *$λ$*, *and $k$ is called the ** wave number**. Therefore

$$k = 2π/λ$$

and it has dimensionality $[k] =$ 1/L. Choosing $k$ selects how fast the wave will be changing in space.

4. *What's *$ω$? This was put in to give correct units to what happens when time changes. So, as in part 3, this time weconsider a fixed $x$ position, say for convenience $x = 0$. When we look at the time variation of a particular bead, we get something proportional to $\sin{ωt}$. It looks like the figure below.

We've put a little clock to indicate that the axis is a time measurement.

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 the argument of sin, $ωt$ changes by $2π$, then $t$ must change by $2π/ω$. Therefore, when $t$* *changes by $2π/ω$*,* the sine 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 including the units of the angles. 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 parameters $ω$ and $k$, but we have a relationship between them: $ω = kv_0$. 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

$$ω = kv_0$$

Expressing $\omega$ and $k$ in terms of $f$ and $\lambda$, we get

$$(2πf) = (2π/λ)v_0$$

so the $2\pi$s cancel and we can get

$$fλ = v_0.$$

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

$$λ = v_0T.$$

This relation makes good sense: If we wiggle a string in our hand to generate a wave, in the time we go through one full oscillation (the period, $T$) a full wiggle will have run out onto the string. In that time the start of the wiggle has already moved along the string for a time $T$. Since the wiggle is moving with a speed $v_0$, the wavelength, $λ$, must be just $v_0T$*.*

Joe Redish 3/31/12* , *Wolfgang Losert 4/6/13* *

Last Modified: March 30, 2022