The wave function of sinusoidal waves
Wave function
The wave function of a wave is a mathematical tool to quantify the changes in the state of the wave over time and at different locations. Since both time and position affect this state, the wave function is a function of two variables: the time \(t\) and the position \(\mathbf{r}.\) For simplicity, we shall consider one-dimensional wave processes, where the position can be simplified to a single coordinate \(x.\)
The wave function \(y(x, t)\) may represent a transverse displacement \(y\) from the equilibrium position for transverse mechanical waves such as waves on a rope, or a longitudinally varying quantity such as density or pressure for longitudinal mechanical waves; for electromagnetic waves, it quantifies the magnitude of the electric or the magnetic field. In all cases, it is a function of both time and position.
Evolution of the wave function
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Let us consider a rope wave travelling to the right, as depicted in the figure. In this case, the wave function \(y(x, t)\) represents the deflection of the rope as measured from the taut, straight position, and in wave motion, it varies in time and also varies by position. We can interpret the picture to the left as two snapshots of the rope wave superposed on each other: the earlier state (at time \(t = 0\)) in grey, the later state (at time \(t\)) in orange. As we can see in the figure, at any position \(x\) we pick at time \(t,\) the wave function \(y\) assumes the same value it had at the beginning (at \(t = 0\)) at a location which is to the left of the position we picked by an amount equal to the distance the wave has travelled in time \(t,\) that is, \(v t.\) This means that the wave function at the selected position \(x\) at the later time \(t\) is equal to the initial value of the wave function (that is, its value at \(t = 0\)) at the position \(v t\) to the left of \(x,\) that is, \(x - vt\): \[y(x, t) = y(x - vt, 0).\] |
We can generalise this observation for any type of wave. If the wave travels to the positive \(x\) direction, the wave function \(y(x, t)\) at any position \(x\) and any time \(t\) can be traced back to the initial wave function \(y(x, 0)\) evaluated with a negative shift \(-v t\) in the coordinate:
\[y(x, t) = y(x - vt, 0).\]
If the wave travels in the negative \(x\) direction, the position is shifted in the positive direction:
\[y(x, t) = y(x + vt, 0).\]
The wave function of sinusoidal waves
A sinusoidal wave is one whose wave function is proportional to a sine at any time \(t:\)
\[y(x, t) = A \cdot \sin\left(k x - \omega t\right) = A \cdot \sin\left(2 \pi \left[\frac{x}{\lambda} - \frac{t}{T} \right]\right)\]
for a wave travelling in the positive \(x\) direction, and
\[y(x, t) = A \cdot \sin\left(k x + \omega t\right) = A \cdot \sin\left(2 \pi \left[\frac{x}{\lambda} + \frac{t}{T} \right]\right)\]
for a wave travelling in the negative \(x\) direction. In these formulae, \(A\) denotes the amplitude, \(\lambda\) stands for the wavelength and \(T\) is the period.
Obtaining the wave function of sinusoidal waves
In what follows, we shall show how to find the formulae for the wave function listed above. We have seen that the wave function can be traced back to the initial value, so first we need not deal with the time dependence, only with the position dependence. The maximum of the wave function must be the amplitude \(A\) of the wave, so the sine function must be multiplied by the amplitude. The argument of the sine function must be an angle, so we cannot use the position \(x\) directly, but with a coefficient \(a\) that transforms it into an angle. The initial wave function, like the wave function at all times, must be periodic in space with \(\lambda,\) whilst the sine function is periodic with \(2 \pi\):
\[A \sin\left(a [x + \lambda]\right) = A \sin\left(a x\right), \Rightarrow a(x + \lambda) = a x + 2 \pi,\]
which means that the unknown coefficient \(a\) we introduced is none other but the (angular) wave number \(k\):
\[a x + a \lambda = a x + 2 \pi, \Rightarrow a = \frac{2 \pi}{\lambda} = k.\]
The initial waveform is thus
\[y(x, 0) = A \sin\left(kx\right).\]
Using the train of thought above, we can obtain the wave function at any time from the initial expression (for a wave travelling in the positive \(x\) direction):
\[y(x, t) = y(x - vt, 0) = A \sin\left(k [x - vt]\right) = A \sin\left(k x - k v t\right) = A \sin\left(k x - k \frac{\omega}{k} t\right) = A \sin\left(k x - \omega t\right).\]
Using the definition of the (angular) wave number \(k\) and of the angular frequency \(\omega,\) we can see that the argument of the sine is an expression that combines the position in wavelength units with the time in period units:
\[y(x, t) = A \sin\left(k x - \omega t\right) = A \sin\left(\frac{2 \pi}{\lambda} x - \frac{2 \pi}{T} t\right) = A \sin\left(2 \pi\left[\frac{x}{\lambda} - \frac{t}{T}\right]\right).\]