One of the common initial conditions given for the wave equation,

$$\frac{\partial^2 p}{\partial t^2} - \nabla^2 p = 0,$$

is $p(\overline{x},t=0) =0$ and $p^\prime (\overline{x},t=0) =0$. What is the physical interpretation of the initial condition $p^\prime (\overline{x}, t=0) =0$?

Edit 1: I messed up the initial conditions. They are now fixed.

Edit 2: Since someone asked, it is an acoustic wave.

  • $\begingroup$ Really? Both at the same time at all locations? Can you please give a reference. Also the "interpretation" in a physical sense may be better explained if the specific type of wave was mentioned, i.e. acoustic, electromagnetic, quantum mechanical, etc. $\endgroup$ – ggcg Jun 17 at 17:54
  • $\begingroup$ @ggcg Sorry, there is a major typo. It is an acoustic wave. $\endgroup$ – midget_messiah Jun 18 at 7:31
  • $\begingroup$ In that case the derivative of the pressure field is proportional to the local displacement or velocity field. $\endgroup$ – ggcg Jun 18 at 12:01
  • $\begingroup$ Welcome to Physics! SE posts are version controlled, so please do not make your post look like a revision table, instead just seamlessly integrate the new material into the post. There is an edit history button at the bottom of the post for those interested in seeing what changed. $\endgroup$ – Kyle Kanos Jun 18 at 13:07

Just like in describing the trajectory of an object under a constant force (like an apple falling under gravity), we have a differential equation describing how it moves $F=ma=-mg$ but the full motion is given by $x(t) = x_0 + x'_0 t -\frac{1}{2}gt^2$. The fact this is a second order differential equation means that you need 2 initial parameters, the initial position and the initial velocity.

Now for the string each point on the string has its own second order differential equation so each point needs an initial position and an initial velocity to describe fully determine the motion.

As a side note I think you should check he initial conditions that you give, because if the string is flat in the beginning, and it is not moving, then it will not move. A far more common initial condition would be something like $p_1(x,t=0) = \text{sin}(x),$ and $p_1'(x,t=0) = 0$. This would describe a string in an initial sin wave shape, with no initial velocity.

Similarly you could have $p_2(x,t=0) = 0,$ and $p_2'(x,t=0) = \text{sin}(x)$. This would be a flat string with initial velocity distributed like a sine wave. These two problems are actually related to each other by a time shift of $\Delta t = -\frac{\pi}{2}$, with corresponding solutions of $p_1(x,t) = \text{sin}(x)\text{cos}(t)$, and $p_2(x,t) = \text{sin}(x)\text{sin}(t)$

Thanks to @MichaelSeifert for the insightful addition in the comments (which I add here). My solutions are specifically for a string that has both ends tied down at $p(0,t) = p(\pi,t) = 0$. Therefore no disturbances are entering from the outside, and the string will not start moving. If instead we had one side was waving up and down in time, $p(0,t) = f(t)$ this would no longer be the appropriate motion. A good example of this kind of phenomena would be a speaker on one side of an open tube, it is driving the oscillations at $x = 0$ which then disperse over the medium. In this case even if the string was not moving and was flat it would start to move because of the disturbances that are entering into the region of interest.

I say string because that is the simplest way for me to visualize 1 dimensional oscillations, but if this were in 3-dimensions it would just as well explain sound waves in air.

  • $\begingroup$ Do you mean $p(x,t=0)=0$ and $p'(x,t=0)=sinx$? $\endgroup$ – orion Jun 17 at 17:13
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    $\begingroup$ The conditions I gave would describe a string in an initial shape of a sine wave, but none of those points have an initial velocity. However the forces between the points will still make it oscillate. You describe a different (but equally valid) initial setup of a flat string where each of the points have velocities according to a sin wave. These are simply related to each other by a time shift of $\frac{\pi}{2}$ and have solutions $p = \text{sin}(x) \text{cos}(t)$, vs $p = \text{sin}(x) \text{sin}(t)$ $\endgroup$ – TEH Jun 17 at 17:39
  • $\begingroup$ In the answer and comment I am ignoring things like length of the string, boundary conditions, wavelengths, and frequencies and simply focusing on the interpretations of the initial conditions. This leads to the equations having inconsistent units. $\endgroup$ – TEH Jun 17 at 17:42
  • $\begingroup$ $F = ma = -mg\ $ instead of $\ F=ma=-g\ $ $\endgroup$ – MarianD Jun 17 at 18:08
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    $\begingroup$ Even if $p(x,0) = \dot{p}(x,0) = 0$, you can still have $p \neq 0$ at later times depending on the spatial boundary conditions. For example, having $p(0, t) = f(t)$ and $p(L,t) = 0$ would be a way of describing sound waves coming from a speaker at $x = 0$ and propagating through a tube with an open end. $\endgroup$ – Michael Seifert Jun 18 at 13:59

Consider a string of length $L$ and mass $M$, and homogeneous density $\lambda = M / L$ and tension $F$, obeying the wave equation \begin{equation}\tag{1} \frac{1}{v^2} \, \frac{\partial^2 y}{\partial t^2} - \frac{\partial^2 y}{\partial x^2} = 0, \end{equation} where $v = \sqrt{F/\lambda}$ is the wave velocity on the string. The general solution to (1) is this: \begin{equation}\tag{2} y(x, t) = \frac{1}{2} \big(\mathcal{Y}(x - v t) + \mathcal{Y}(x + v t) \big) + \frac{1}{2 v} \int_{x - v t}^{x + v t} \mathcal{V}(u) \, du, \end{equation} where $\mathcal{Y}(x) \equiv y(x, 0)$ is the initial displacement of the string (for each $x$) and $\mathcal{V}(x) \equiv \dot{y}(x, 0)$ is the initial velocity of each of its elements. $\mathcal{Y}(x)$ and $\mathcal{V}(x)$ are arbitrary continuous and derivable functions.

If the string is fixed at $x = 0$: $y(0, t) = 0$, then $\mathcal{Y}(x)$ and $\mathcal{V}(x)$ must be odd functions: $\mathcal{Y}(-v t) = - \mathcal{Y}(v t)$. For example: $\mathcal{Y}(x) = A \sin{(k \, x)}$ and $\mathcal{V}(x) = 0$ (no velocity at time $t = 0$). If the string is also fixed at the other side: $y(L, t) = 0$, then $k = n \pi / L$ where $n = 1, 2, 3, \ldots, \infty$.

You may also have a flat string at $t = 0$: $\mathcal{Y}(x) = 0$, but give an initial velocity to each of its elements: $\mathcal{V}(x) = B \sin{(k \, x)}$ (with $k = n \pi / L$ so the string stay fixed at both of its ends: $\mathcal{V}(0) = \mathcal{V}(L) = 0$).


To answer your question "What is the physical interpretation of the initial condition $p^\prime (\overline{x}, t=0) =0$ ".

$p^\prime (\overline{x}, t=0) $ is the initial velocity at time zero as a function of $\overline{x}$ .

The first initial condition, $p (\overline{x}, t=0) $, is the initial displacement at time zero as a function of $\overline{x}$ . Note that the initial conditions can be given independently of each other. The initial displacement can be zero with a non zero initial velocity. If both are zero there will be no subsequent wave (if there is no driving force).


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