A wave propagates on a string in the positive x-direction at a velocity v. The s

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A wave propagates on a string in the positive x-direction at a velocity v. The shape of the string at

t = t_{0}

is given by

g (x, t_{0})

= A sin

(\frac{x}{a})

Write the wave equation for a general time t.

Options:

A . A sin[xa−v(t−t0)]

B . A sin[x−v(t−t0)a]

C . A sin[x+v(t−t0)a]

D . None of these

Answer: Option B
:
B
Here given is

g (x, t_{0}) = A s i n (\frac{x}{a})

at

t = t_{0}

Comparing with the general equation g(x,t)=

A s i n (k x - ω t + ϕ)

We get at

t = t_{0}

k = \frac{1}{a}

also

- ω t_{0} + ϕ = 0

\Rightarrow ϕ = ω t_{0}

We also known that

k = \frac{ω}{v}

\Rightarrow ω = \frac{v}{a}

so we get

g (x, t) = A s i n (\frac{x}{a} - \frac{v t}{a} + \frac{v t_{0}}{a})

g (x, t) = A s i n [\frac{x}{a} - \frac{v t}{a} + \frac{v t_{0}}{a}]

= A s i n (\frac{x - v (t - t_{0})}{a})

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