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Good day! I'm having trouble comprehending this question.

Write the equation for a standing wave that has three antinodes of amplitude 2.00 cm on a 3.00 m long string that is fixed at both ends and vibrates 15.0 times a second. The time t = 0 is chosen to be an instant when the string is flat. If a wave pulse was propagated along this string, how fast would it travel?

The equation pertains to $y(x,t)=A_{SW}sin(kx)sin(\omega t)$, right? That's the equation for standing waves that is fixed at both ends.

Now, plugging everything in gives $y(x,t)=(0.04 m)sin[(100\pi rad/m)x]sin[(30\pi rad/s)t]$ I understand that, to get the transverse velocity, I need to get the partial derivative of the function in terms of $t$. The velocity is $\frac{6}{5}\pi m/s$.

However, when I read the question again, it seems that it asks for the speed of the wave since the frequency and length are given. Since the entire wave has three antinodes, the $\lambda $ should be $\frac{2}{3}L$, and I can use $v=f\lambda$ from there. My answer using this approach is 30 m/s.

Which is the correct approach?

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A traveling wave has a frequency determined by its source and a speed determined by the properties of the medium. Particles (or fields) in the medium may have a displacement or velocity from a zero value described by a sine (or cosine) function of position and time. If the wave is bouncing back and forth between boundaries it can resonate at certain frequencies (if the boundary conditions are met) and produce a standing wave. Note that the speed of the traveling wave is not the same as the speed of the particles in the medium. Your standing wave function describes the motion of the particles. The amplitude is given as 2 cm. (not .o4m). With 3 anti-nodes within 3 m, the wavelength is 2 m. The k = 2π/λ = π. Were it correct, the speed that you calculated would be a maximum for a particle at the center of an anti-node.

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