Question

Hello, I am wondering how I can determine the direction in which a wave is travelling given a function I must verify if it first is a solution to the wave equation by finding the partial derivatives then I do not know how to extrapolate the direction and speed of a wave. Thanks

Answer 1

We will discuss the case of 1-D waves (like waves travelling along a string). The function given to you will be a function of position (x) and time (t). The function will be like y = f(x,t).

To verify if it is actually a wave equation or not:

First calculate and   .

Now if we can write,    where c is some constant (actually square of speed of wave) then the given function is actually representing wave equation.

In fact any function of where a and b are constants will represent a wave equation if it is finite at every point at every time.

E.g. y = A sin (ax + bt)

  y = A cos (ax - bt)

  

all these represent wave equations.

To find the direction of propagation of wave:

This is very simple. In the given equation, if coefficient of x and t have same sign, the wave is travelling in negative direction and if they have opposite signs, the wave is travelling in positive direction.

E.g. y = 5 sin (3x - 4t) travels in +ve direction as coefficients of x (3) and t (-4) have opposite signs.

y = 5 sin (-3x + 4t) travels in +ve direction as coefficients of x (-3) and t (4) have opposite signs.

y = 5 sin (3x + 4t) travels in -ve direction as coefficients of x (3) and t (4) have same signs.

y = 5 sin (-3x - 4t) travels in -ve direction as coefficients of x (-3) and t (-4) have opposite signs.