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Here is the equation of a wave

(1) $\,y(x,t)=2\sin(4x-2t)$

What is the wave speed?

What is the maximum speed perpendicular to the wave's direction of travel (transverse speed)?

I understand both questions and know how to determine the wave speed but the textbook never mentioned anything about transverse speed until now.

Can someone please explain what exactly transverse speed is in detail and the derivation of the formula? The only thing that comes to mind is the velocity of SHM since it is basically asking how fast the particle or element is moving vertically as the wave propagates.

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The transverse speed is the rate at which the orthogonal components of the wave move.

Treat the derivation of the transverse speed equation as SHM, as you mention in your post.

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Consider a medium in which the motion of the particles is described by the given wave equation,

The given equation basically describes an SHM which moves forward. If you consider the particles at x=0, you can easily see that it is 2sin(-2t). As for a normal SHM, the maximum velocity for the particle occurs at the mean position and is 4 units/second. This particular particle at x=0 is moving up and down and its speed is the transverse speed.

Now consider all the particles describing the wave. The speed with which this locus moves forward is the wave speed. This is the distance between two consecutive crests or troughs of the wave i.e the wavelength travelled in the time period of the wave i.e the time taken for the wave to traverse one wavelength. The time period you can easily get from SHM as 2π/T=omega=2 in this equation. For wavelength, since the equation is a sine function, by considering a particular instant t=t', we get it from the equation as follows,

4(x+ λ)=4x+2π which implies that 2π/λ=4.Therefore λ/T=2/4=0.5, which is the required wave speed. A useful lesson that you can learn from this for future reference is that for a wave equation, dividing coefficient of t( time variable) by coefficient of x( position variable) always gives wave speed.

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