# How do I plot ψ(x,t) as a function of x at time t=a/v?

• whatisreality
In summary, the 1D wave function ψ(x,t) satisfies initial conditions of ψ(x,0) = 0 for all x and ∂ψ/∂t (x,0) = v for -a≤x≤a and 0 otherwise. Using d'Alembert's formula, the function simplifies to ψ(x,t) = 1/2v ∫∂ψ/∂t(x,0) dx. To plot the graph at t=a/v, the integral must be evaluated with respect to x, using the given initial condition of v. The resulting function is valid for all x in [-a,a] and can be plotted for any given value
whatisreality

## Homework Statement

A 1D wave function ψ(x,t) satisfies these initial conditions:
ψ(x,0) = 0 for all x

∂ψ/∂t (x,0) is v for -a≤x≤a
0 otherwise
Plot ψ(x,t) as a function of x at time t=a/v.

## The Attempt at a Solution

I know the 1D wave equation is given by d'Alembert's:
##\psi(x,t) = 0.5[\psi(x+vt,0) + \psi(x-vt,0)] + \frac{1}{2v} \int_{x-vt}^{x+vt} \frac{\partial \psi}{\partial t}(x,0) \mathrm dx##

But for this function because ψ(x, 0) = 0 for all x, that simplifies to
##\psi(x,t) = \frac{1}{2v} \int_{x-vt}^{x+vt} \frac{\partial \psi}{\partial t}(x,0) \mathrm dx##

I wanted to sketch ψ(x,t) first but I'm not sure how to evaluate the integral. For -a≤x≤a,
##\frac{\partial \psi}{\partial t}(x,0)## = v,
But do I still integrate between x+vt and x-vt? Do I sub x=a or x=-a into the limits, or maybe t=0? I;m fairly sure you just integrate v between x+vt and x-vt (w.r.t x).

Then I have to plot the graph at t=a/v. And I'm not given what
##\frac{\partial \psi}{\partial t}(x,\frac{a}{v})## is, and there's no easy relationship I can spot between the two graphs in terms of for example ψ(x,0) is ψ(x, a/v) translated or rotated etc.

In fact, I can't see how it being t=a/v instead of t=0 would affect the graph at all!

Last edited:
When you do the integral, you will have a function ##\psi(x,t)## which is of x and t and is valid in [-a,a].
Then you can find what that function is for any particular time t ... which will be a function of x alone.

## What is a wave equation?

A wave equation is a mathematical formula that describes the behavior of a wave as it propagates through a medium. It is often used to model physical phenomena such as sound waves, light waves, and water waves.

## How do you sketch a graph of a wave equation?

To sketch a graph of a wave equation, you first need to identify the variables involved, such as amplitude, wavelength, and frequency. Then, use the wave equation to calculate the y-values for different x-values. These y-values can be plotted on a graph to create a visual representation of the wave's behavior.

## What is the significance of the amplitude on a wave graph?

The amplitude on a wave graph represents the maximum displacement of the wave from its equilibrium position. It is a measure of the wave's intensity or strength, with larger amplitudes indicating a more powerful wave.

## How does the wavelength affect a wave graph?

The wavelength on a wave graph is the distance between two consecutive peaks or troughs of the wave. It determines the distance traveled by the wave in one complete cycle and is inversely proportional to the frequency of the wave. Longer wavelengths correspond to lower frequencies and vice versa.

## What is the relationship between frequency and period in a wave equation?

Frequency and period are inversely related in a wave equation. Frequency is the number of complete cycles the wave completes in one second, while period is the time it takes for the wave to complete one full cycle. The higher the frequency, the shorter the period, and vice versa.

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