A human wave. During sporting events within large, densely packed stadiums, spectators will send a wave (or pulse) around the stadium (in the figure shown below). As the wave reaches a group of spectators, they stand with a cheer and then sit. At any instant, the width $\mathrm{w}$ of the wave is the distance from the leading edge (people are just about to stand) to the trailing edge (people have just sat down). Suppose a human wave travels a distance of 853 seats around a stadium in $51 \mathrm{s}$, with spectators requiring about $1.8 \mathrm{s}$ to respond to the wave's passage by standing and then sitting. What are (a) the wave speed $\mathrm{v}$ (in seats per second) and (b) width w (in number of seats)?

Use the wave equation to find the speed of a wave given by
$y\mathit{(}x\mathit{,}t)=(2.00 \mathrm{mm}){\left[\left(15.0 {\mathrm{m}}^{-1}\right)x-\left(8.00 \mathrm{rad} {\mathrm{s}}^{-1}\right)t\right]}^{0.5}$

In the figure shown below a string, tied to a sinusoidal oscillator at $P$ and running over a support at $Q,$ is stretched by a block of mass $m$ Separation $L=1.20 \mathrm{m},$ linear density $\text{μ}=1.20 \mathrm{g} {\mathrm{m}}^{-1},$ and the oscillator frequency $f=120\mathrm{Hz}$. The amplitude of the motion at $P$ is small enough for that point to be considered a node. A node also exists at $Q$. (a) What mass $m$ allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any can be set up if $m=1.00 \mathrm{kg}?$

In the figure shown below, a sinusoidal wave moving along a string is shown twice as crest $A$ travels in the positive direction of an $x$ axis by distance$d=6.0\mathrm{cm}$ in  $3.0\mathrm{ms}$. The tick marks along the axis are separated by $10\mathrm{cm};$ height $\mathrm{H}=6.00\mathrm{mm}.$ The equation for the wave is in the form$y(x\mathit{,}t)=y_{\mathrm{m}}\sin (kx\pm \omega t),$ so what are $\left(\mathrm{a}\right)y_m,\left(\mathrm{b}\right)k,\left(\mathrm{c}\right)\omega ,$ and
(d) the correct choice of sign in front of $\omega ?$

In the figure shown below a string, tied to a sinusoidal oscillator at $P$ and running over a support at $Q$ , is stretched by a block of mass $m$. The separation $L$ between $P$ and $Q$ is $1.20 \mathrm{m},$ and the frequency $f$ of the oscillator is fixed at $120 \mathrm{Hz}$. The amplitude of the motion at $P$ is small enough for that point to be considered a node. A node also exists at $Q.$ A standing wave appears when the mass of the hanging block is $286.1 \mathrm{g}$ or $447.0 \mathrm{g}$ , but not for any intermediate mass. What is the linear density of the string?

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