Wave Equation Calculator

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Example

Created on 2024-06-20Asked by Owen Flores (Solvelet student)

Find the solution to the one-dimensional wave equation

u_{tt} = c^2 u_{xx}

subject to the initial conditions

u(x, 0) = f(x)

and

u_t(x, 0) = g(x)

, where

c

is the wave speed and

f(x)

and

g(x)

are given functions.

Solution

To solve the one-dimensional wave equation

u_{tt} = c^2 u_{xx}

subject to the initial conditions

u(x, 0) = f(x)

and

u_t(x, 0) = g(x)

, we use the method of separation of variables. 1. **Solution by separation of variables:** We assume a solution of the form

u(x, t) = X(x)T(t)

. Substituting this into the wave equation, we get:

X(x)T''(t) = c^2X''(x)T(t).

2. **Separating variables:** Dividing both sides by

c^2X(x)T(t)

, we get:

\frac{T''(t)}{c^2T(t)} = \frac{X''(x)}{X(x)}.

This results in two ordinary differential equations:

T''(t) + c^2k^2T(t) = 0,

X''(x) + k^2X(x) = 0,

where

k

is a separation constant. 3. **Solving the ODEs:** - The solution to the time-dependent equation is

T(t) = A\cos(ckt) + B\sin(ckt)

, where

A

and

B

are constants. - The solution to the space-dependent equation is

X(x) = C\cos(kx) + D\sin(kx)

, where

C

and

D

are constants. 4. **Combining solutions:** The general solution to the wave equation is a linear combination of the solutions to the time-dependent and space-dependent equations:

u(x, t) = \sum_{n=1}^{\infty} (A_n\cos(k_nx) + B_n\sin(k_nt))(C_n\cos(ck_nt) + D_n\sin(ck_nt)),

where

k_n = \frac{n\pi}{L}

for

n = 1, 2, 3, \ldots

and

L

is the length of the interval. 5. **Applying initial conditions:** Use the given initial conditions

u(x, 0) = f(x)

and

u_t(x, 0) = g(x)

to determine the coefficients

A_n

and

B_n

. 6. **Result:** The solution to the one-dimensional wave equation subject to the given initial conditions is given by the expression obtained in step 4. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Abigail Young on Solvelet

1. Find the solution to the one-dimensional wave equation

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

subject to boundary conditions.2. Determine the wave speed of a wave described by the equation

u(x,t) = 3\sin(2x - 4t)

DefinitionA wave equation is a second-order linear partial differential equation for the description of waves-as they occur in physics-such as sound waves, light waves and water waves. In its full form, it is written as ∂t2∂2u=c2∇2u, where u is the wave function and c is the wave speed. Example: The wave equation gives the solutions as oscillations, such as a vibrating string or when discussing electromagnetic waves

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