General Solution of 1D Wave Equation

Consider the one-dimensional wave equation,

$$\frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\,\frac{\partial^{\,2}\psi}{\partial x^{\,2}},$$ (8.33)

where $\psi(x,t)$ is the wavefunction, and $v$ the characteristic phase velocity. We have seen a number of particular solutions of this equation. For instance,

$$\psi(x,t) = A\,\cos(\omega\,t-k\,x-\phi)$$ (8.34)

represents a traveling wave of amplitude $A$, angular frequency $\omega$, wavenumber $k$, and phase angle $\phi$, that propagates in the positive $x$-direction. The previous expression is a solution of the one-dimensional wave equation, (8.33), provided that it satisfies the dispersion relation

$$\omega = k\,v;$$ (8.35)

that is, provided the wave propagates at the fixed phase velocity $v$. We can also write the wavefunction (8.34) in the form

$$\psi(x,t) = C_+\,\cos[k\,(v\,t-x)] + S_+\,\sin[k\,(v\,t-x)],$$ (8.36)

where $C_+=A\,\cos\phi$, $S_+=A\,\sin\phi$, and we have explicitly incorporated the dispersion relation $\omega=k\,v$ into the solution. The previous expression can be regarded as the most general form for a traveling wave of wavenumber $k$ propagating in the positive $x$-direction. Likewise, the most general form for a traveling wave of wavenumber $k$ propagating in the negative $x$-direction is

$$\psi(x,t) = C_-\,\cos[k\,(v\,t+x)] + S_-\,\sin[k\,(v\,t+x)].$$ (8.37)

We have also encountered standing wave solutions of Equation (8.33). However, as we have seen, these can be regarded as linear superpositions of traveling waves, of equal amplitude and wavenumber, propagating in opposite directions. (See Section 6.4.) In other words, standing waves are not fundamentally different to traveling waves.

The wave equation, (8.33), is linear. This suggests that its most general solution can be written as a linear superposition of all of its valid wavelike solutions. In the absence of specific boundary conditions, there is no restriction on the possible wavenumbers of such solutions. Thus, it is plausible that the most general solution of Equation (8.33) can be written

$$\psi(x,t) = \int_{-\infty}^{\infty} C_+(k)\,\cos[k\,(v\,t-x)]\,dk+ \int_{-\infty}^{\infty} S_+(k)\,\sin[k\,(v\,t-x)]\,dk$$

$$~~~~+ \int_{-\infty}^{\infty} C_-(k)\,\cos[k\,(v\,t+x)]\,dk+ \int_{-\infty}^{\infty} S_-(k)\,\sin[k\,(v\,t+x)]\,dk.$$ (8.38)

In other words, the general solution is a linear superposition of traveling waves propagating to the right (i.e., in the positive $x$-direction), and to the left. Here, $C_+(k)$ represents the amplitude of right-propagating cosine waves of wavenumber $k$ in this superposition. Moreover, $S_+(k)$ represents the amplitude of right-propagating sine waves of wavenumber $k$, $C_-(k)$ the amplitude of left-propagating cosine waves, and $S_-(k)$ the amplitude of left-propagating sine waves. Because each of these waves is individually a solution of Equation (8.33), we are guaranteed, from the linear nature of this equation, that the previous superposition is also a solution.

But, how can we prove that Equation (8.38) is the most general solution of the wave equation, (8.33)? Our understanding of Newtonian dynamics tells us that if we know the initial wave amplitude $\psi(x,0)$, and its time derivative $\dot{\psi}(x,0)$, then this should constitute sufficient information to uniquely specify the solution at all subsequent times. Hence, if Equation (8.38) is the most general solution of Equation (8.33) then it must be consistent with any initial wave amplitude, and any initial wave velocity. In other words, given any $\psi(x,0)$ and $\dot{\psi}(x,0)$, we should be able to uniquely determine the functions $C_+(k)$, $S_+(k)$, $C_-(k)$, and $S_-(k)$ appearing in Equation (8.38). Let us see if this is the case.

From Equation (8.38),

$$\psi(x,0) = \int_{-\infty}^{\infty}\left[C_+(k)+C_-(k)\right]\,\cos(k\,x)\,dk+ \int_{-\infty}^{\infty}\left[-S_+(k)+S_-(k)\right]\,\sin(k\,x)\,dk.$$ (8.39)

However, this is a Fourier transform of the form (8.10). Moreover, Equations (8.11) and (8.12) allow us to uniquely invert this transform. In fact,

$$C_+(k) + C_-(k) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \psi(x,0)\,\cos(k\,x)\,dx,$$ (8.40)

$$-S_+(k) + S_-(k) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \psi(x,0)\,\sin(k\,x)\,dx.$$ (8.41)

Equation (8.38) also yields

$$\dot{\psi}(x,0) = \int_{-\infty}^{\infty}k\,v\left[C_+(k)-C_-(k)\... ...,x)\,dk+ \int_{-\infty}^{\infty}k\,v\left[S_+(k)+S_-(k)\right]\,\cos(k\,x)\,dk.$$ (8.42)

This is, again, a Fourier transform, and can be inverted to give

$$k\,v\,[C_+(k) - C_-(k)] = \frac{1}{2\pi}\int_{-\infty}^{\infty} \dot{\psi}(x,0)\,\sin(k\,x)\,dx,$$ (8.43)

$$k\,v\,[S_+(k) + S_-(k)] = \frac{1}{2\pi}\int_{-\infty}^{\infty} \dot{\psi}(x,0)\,\cos(k\,x)\,dx.$$ (8.44)

Hence,

$$C_+(k) =\frac{1}{4\pi}\left[\int_{-\infty}^{\infty}\psi(x,0)\,\cos(k\,x)\,dx + \int_{-\infty}^\infty \frac{\dot{\psi}(x,0)}{k\,v}\,\sin(k\,x)\,dx\right],$$ (8.45)

$$C_-(k) =\frac{1}{4\pi}\left[\int_{-\infty}^{\infty}\psi(x,0)\,\cos(k\,x)\,dx - \int_{-\infty}^\infty \frac{\dot{\psi}(x,0)}{k\,v}\,\sin(k\,x)\,dx\right],$$ (8.46)

$$S_+(k) =\frac{1}{4\pi}\left[-\int_{-\infty}^{\infty}\psi(x,0)\,\sin(k\,x)\,dx + \int_{-\infty}^\infty \frac{\dot{\psi}(x,0)}{k\,v}\,\cos(k\,x)\,dx\right],$$ (8.47)

$$S_-(k) =\frac{1}{4\pi}\left[\int_{-\infty}^{\infty}\psi(x,0)\,\sin(k\,x)\,dx +\int_{-\infty}^\infty \frac{\dot{\psi}(x,0)}{k\,v}\,\cos(k\,x)\,dx\right].$$ (8.48)

It follows that we can uniquely determine the functions $C_+(k)$, $C_-(k)$, $S_+(k)$, and $S_-(k)$, appearing in Equation (8.38), for any $\psi(x,0)$ and $\dot{\psi}(x,0)$. This proves that Equation (8.38) is indeed the most general solution of the wave equation, (8.33).

Let us examine our solution in more detail. Equation (8.38) can be written

$$\psi(x,t) = F(v\,t-x)+ G(v\,t+x),$$ (8.49)

where

$$F(x) = \int_{-\infty}^\infty\left[C_+(k)\,\cos(k\,x)+ S_+(k)\,\sin(k\,x)\right] dk,$$ (8.50)

$$G(x) = \int_{-\infty}^\infty\left[C_-(k)\,\cos(k\,x)+ S_-(k)\,\sin(k\,x)\right] dk.$$ (8.51)

(See Exercise 6.) What is the significance of Equation (8.49)? Actually, $F(v\,t-x)$ represents a wave disturbance of arbitrary shape that propagates in the positive $x$-direction, at the fixed speed $v$, without changing shape. This follows because a point with a given amplitude on the wave, $F(v\,t-x)=c$, has an equation of motion $v\,t-x=F^{\,-1}(c)={\rm constant}$, and thus propagates in the positive $x$-direction at the speed $v$. Moreover, because all points on the wave propagate in the same direction at the same speed, it follows that the wave does not change shape as it moves. By analogy, $G(v\,t+x)$ represents a wave disturbance of arbitrary shape that propagates in the negative $x$-direction, at the fixed speed $v$, without changing shape. We conclude that the most general solution to the wave equation, (8.33), is a superposition of two wave disturbances of arbitrary shapes that propagate in opposite directions, at the fixed speed $v$, without changing shape. Such solutions are generally termed wave pulses. What is the relationship between a general wave pulse and the sinusoidal traveling wave solutions to the wave equation that we found previously? As is apparent from Equations (8.50) and (8.51), a wave pulse is a superposition of sinusoidal traveling waves propagating in the same direction as the pulse. Moreover, the amplitude of cosine waves of wavenumber $k$ in this superposition is the cosine Fourier transform of the pulse shape, evaluated at wavenumber $k$. Likewise, the amplitude of sine waves of wavenumber $k$ in the superposition is the sine Fourier transform of the pulse shape, evaluated at wavenumber $k$.

Figure 8.3: Fourier transform of a triangular wave pulse.

For instance, suppose that we have a triangular wave pulse of the form

$$F(x) = \left\{ \begin{array}{lcc} 1 -2\,\vert x\vert/l&\mbox{... ...vert\leq l/2\\ [0.5ex] 0 &&\vert x\vert>l/2 \end{array}\right..$$ (8.52)

The sine Fourier transform of this pulse shape is zero by symmetry. However, the cosine Fourier transform is

$$C(k)=\frac{1}{2\pi}\int_{-\infty}^\infty F(x)\,\cos(k\,x)\,dx = \frac{l}{4\pi}\,\frac{\sin^2(k\,l/4)}{(k\,l/4)^{\,2}}.$$ (8.53)

(See Exercise 7.) The functions $F(x)$ and $C(k)$ are shown in Figure 8.3. It follows that the right-propagating triangular wave pulse

$$\psi(x,t)= \left\{ \begin{array}{lcc} 1 -2\,\vert v\,t-x\vert... ...\leq l/2\\ [0.5ex] 0 &&\vert v\,t-x\vert>l/2 \end{array}\right.$$ (8.54)

can be written as the following superposition of right-propagating cosine waves:

$$\psi(x,t) =\frac{1}{4\pi} \int_{-\infty}^\infty \frac{\sin^2(k\,l/4)}{(k\,l/4)^{\,2}}\,\cos[k\,(v\,t-x)]\,l\,dk.$$ (8.55)

Likewise, the left-propagating triangular wave pulse

$$\psi(x,t)= \left\{ \begin{array}{ccc} 1 -2\,\vert v\,t+x\vert... ...\leq l/2\\ [0.5ex] 0 &&\vert v\,t+x\vert>l/2 \end{array}\right.$$ (8.56)

becomes

$$\psi(x,t) =\frac{1}{4\pi} \int_{-\infty}^\infty \frac{\sin^2(k\,l/4)}{(k\,l/4)^{\,2}}\,\cos[k\,(v\,t+x)]\,l\,dk.$$ (8.57)

The ideas developed in this section can be extended to multi-dimensional waves in a straightforward fashion. (See Exercises 12 and 13.)