Question

The displacement of the particle at $x=0$ of a stretched string carrying a wave in the positive x-direction is given by $f\left(t\right)=A\sin (t/T)$. The wave speed is $v$. Write the wave equation.

Answer 1

Step 1: Given data:

The displacement of the particle at $x=0$ of a stretched string carrying a wave in the positive x-direction is given by,

At $x=0,f\left(t\right)=A\sin \left(t/T\right)$

Step 2: Formula used:

The wavelength is given by

Wavelength$\left(\lambda \right)=vT$

Where, $v=$speed of the wave and $T=$time period.

The general wave equation is given by

$y=f\left(\left(\frac{t}{T}\right)-\left(\frac{x}{\lambda }\right)\right)\dots \dots \dots \dots \dots \left(1\right)$

Step 3: Calculation for finding wave equation:

The displacement of a particle at $x=0$ of a stretched string carrying a wave in a positive $x-$direction is given by

$f\left(t\right)=A\sin \left(\frac{t}{T}\right)\dots \dots \dots \dots \dots \dots \dots \left(2\right)$

For wave equation having $"v"$ speed

Using equation (1)

$y=f\left(\left(\frac{t}{T}\right)-\left(\frac{x}{\lambda }\right)\right)$

Substituting the value of wavelength as $\left(\lambda \right)=vT$ in equation (1) and from equation (2)

$\begin{array}{rcl}y& =& f\left(\left(\frac{t}{T}\right)-\left(\frac{x}{vT}\right)\right)\\ & =& A\sin \left[\left(\frac{t}{T}\right)-\left(\frac{x}{vT}\right)\right]\end{array}$

Hence the wave equation is $\begin{array}{rcl}f\left(x,t\right)& =& y=A\sin \left(\frac{t}{T}-\frac{x}{VT}\right)\end{array}$.