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We represent the wavefunction of any wave on the string as $$y=f(x-vt),$$ where $v$ is velocity of the wave and $x$ is distance from origin and $t$ is time taken to reach the given point and $y$ is the displacement from the the $x$-axis.

but my question is why do we represent wave by this equation (i have clearly understood that the vertical displacement is the function of distance from the mean origin and time), why can't we represent a wave function by something like $$y=f(x^2-v^2t^2)$$ or any other function and why is there negative sign between $x$ and $vt$. i think negative sign doesn't make any sense.

on referring other book i saw wave equation was represented as $$y=f(t-\frac{x}{v}),$$ are they both same?

Please try to give answer with simple explanation, i am not expert in differentiation or differential equations, so i can't understand their meanings!

Qmechanic
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ramsay
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1 Answers1

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A1: In General, second function is not a solution of wave equation. The general solution is$$y=f(x-vt)+g(x+vt).$$One can show that $$y=f(x^2-v^2 t^2)$$satisfies $x^2 \frac {d^2 y}{dx^2}-t^2\frac {1}{v^2} \frac{d^2 y}{dt^2} $ instead of $\frac {d^2 y}{dx^2}-\frac {1}{v^2} \frac{d^2 y}{dt^2} $ by using Chain rule. Moreover, it doesn't mean propagating wave while $$y=f(x-vt)$$ means propagating wave.
A2: The first representation $$y=f(x-vt)$$ and the third $$y=f (t-\frac {x}{v})$$ is equivalent since it is just a change of variable.

Hyunjin Song
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