The Mathematical Description of a Wave Video Lessons

Concept

# Problem: Which of the following wave functions satisfies the wave equation1. D(x,t) = Acos(kx+ωt)2. D(x,t) = Asin(kx+ωt)3. D(x,t) = A(cos kx + cos ωt)

###### FREE Expert Solution

We can write the given wave equation as:

$\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{\right)}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{v}}^{\mathbf{2}}}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{\right)}}{\mathbf{\partial }{\mathbf{t}}^{\mathbf{2}}}$, where y(x,t) id the wave function and v is the wave velocity.

We differentiate the wave function in option 1. with respect to time by position coordinate x.

Now we have:

$\begin{array}{rcl}\frac{\mathbf{\partial }\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{\right)}}{\mathbf{\partial }\mathbf{x}}& \mathbf{=}& \mathbf{-}\mathbf{k}\mathbf{A}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{k}\mathbf{x}\mathbf{+}\mathbf{\omega }\mathbf{t}\mathbf{\right)}\\ \frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{\right)}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}& \mathbf{=}& \mathbf{-}{\mathbf{k}}^{\mathbf{2}}\mathbf{A}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{k}\mathbf{x}\mathbf{+}\mathbf{\omega }\mathbf{t}\mathbf{\right)}\end{array}$

89% (46 ratings)
###### Problem Details

Which of the following wave functions satisfies the wave equation

$\frac{\partial^2 D}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2D }{\partial t^2}$

1. D(x,t) = Acos(kx+ωt)

2. D(x,t) = Asin(kx+ωt)

3. D(x,t) = A(cos kx + cos ωt)