The linear mass density is $\mu=m/L=2.50/20.0=0.125\,\text{kg m}^{-1}$. The speed of a transverse wave on a stretched string is $v=\sqrt{T/\mu}=\sqrt{200/0.125}=40\,\text{m s}^{-1}$. Time to travel $20.0\,\text{m}$ is $t=L/v=20.0/40=0.50\,\text{s}$.
$0.50\,\text{s}$.
The stone takes $t_1=\sqrt{2h/g}=\sqrt{600/9.8}=7.82\,\text{s}$ to reach the water. The splash sound then takes $t_2=h/v=300/340=0.882\,\text{s}$ to reach the top. Total time $t=t_1+t_2=8.70\,\text{s}$.
$8.7\,\text{s}$ after the stone is dropped.
The linear mass density is $\mu=2.10/12.0=0.175\,\text{kg m}^{-1}$. For a string, $v=\sqrt{T/\mu}$, so $T=\mu v^2=0.175(343)^2=2.06\times10^4\,\text{N}$.
$2.1\times10^4\,\text{N}$ approximately.
For an ideal gas, $P/\rho=RT/M$, where $M$ is molar mass. Substituting in $v=\sqrt{\gamma P/\rho}$ gives $v=\sqrt{\gamma RT/M}$. Thus pressure cancels out at fixed temperature. Increasing temperature increases $T$, so $v$ increases. Moist air contains water vapour whose molar mass is smaller than that of dry air; this reduces the effective molar mass and density for the same pressure and temperature, so the speed increases.
Using the ideal-gas relation, $v=\sqrt{\gamma RT/M}$; hence it is independent of pressure at fixed temperature, increases as $\sqrt{T}$, and increases when humidity lowers the effective molar mass of air.
Each expression is a function of only one travelling combination. In (a), $y=(x-vt)^2$ has the form $f(x-vt)$ and travels in the positive x-direction. In (b), $y=\log[(x+vt)/x_0]$ has the form $f(x+vt)$ and travels in the negative x-direction where the logarithm is defined. In (c), $y=1/(x+vt)$ also has the form $f(x+vt)$, except at its singular point. Thus all can possibly represent travelling-wave shapes.
Yes. All three can represent travelling waves mathematically, subject to their domains.
The frequency remains the same on reflection and transmission: $f=1000\,\text{kHz}=1.0\times10^6\,\text{Hz}$. For reflected sound in air, $\lambda_a=v_a/f=340/(1.0\times10^6)=3.40\times10^{-4}\,\text{m}$. For transmitted sound in water, $\lambda_w=1486/(1.0\times10^6)=1.486\times10^{-3}\,\text{m}$.
(a) $3.40\times10^{-4}\,\text{m}$; (b) $1.486\times10^{-3}\,\text{m}$.
Here $v=1.7\,\text{km s}^{-1}=1700\,\text{m s}^{-1}$ and $f=4.2\,\text{MHz}=4.2\times10^6\,\text{Hz}$. Thus $\lambda=v/f=1700/(4.2\times10^6)=4.05\times10^{-4}\,\text{m}=0.405\,\text{mm}$.
$4.0\times10^{-4}\,\text{m}$, or about $0.40\,\text{mm}$.
The argument has the form $\omega t+kx+\phi$, so the wave travels in the negative x-direction. Here $\omega=36\,\text{s}^{-1}$ and $k=0.018\,\text{cm}^{-1}$. Hence $v=\omega/k=36/0.018=2000\,\text{cm s}^{-1}$. The amplitude is $3.0\,\text{cm}$ and $f=\omega/(2\pi)=36/(2\pi)=5.73\,\text{Hz}$. At $x=0,t=0$, the phase is $\pi/4$. The distance between successive crests is $\lambda=2\pi/k=2\pi/0.018=349\,\text{cm}$.
(a) It is a travelling wave moving in the negative x-direction with speed $2000\,\text{cm s}^{-1}$, or $20\,\text{m s}^{-1}$. (b) Amplitude $3.0\,\text{cm}$ and frequency $5.7\,\text{Hz}$. (c) Initial phase $\pi/4$. (d) $349\,\text{cm}$.
The phase changes with position as $2\pi(0.0080x)$, where $x$ is in cm. Thus $\Delta\phi=2\pi(0.0080\Delta x)$. For $4\,\text{m}=400\,\text{cm}$, $\Delta\phi=2\pi(0.0080)(400)=6.4\pi$. For $0.5\,\text{m}=50\,\text{cm}$, $\Delta\phi=0.8\pi$. Also, by definition, separation $\lambda/2$ gives phase difference $\pi$, and $3\lambda/4$ gives $3\pi/2$.
(a) $6.4\pi\,\text{rad}$; (b) $0.8\pi\,\text{rad}$; (c) $\pi\,\text{rad}$; (d) $3\pi/2\,\text{rad}$.
The form $A\sin kx\cos\omega t$ is a standing wave. Since $2a=0.06$, each travelling component has amplitude $a=0.03\,\text{m}$: $y_1=0.03\sin(2\pi x/3-120\pi t)$ and $y_2=0.03\sin(2\pi x/3+120\pi t)$. Here $k=2\pi/3$, so $\lambda=3.0\,\text{m}$, and $\omega=120\pi$, so $f=60\,\text{Hz}$. The speed is $v=f\lambda=180\,\text{m s}^{-1}$. The linear density is $\mu=3.0\times10^{-2}/1.5=2.0\times10^{-2}\,\text{kg m}^{-1}$. Hence tension $T=\mu v^2=0.020(180)^2=648\,\text{N}$.
(a) A stationary wave. (b) It is the superposition of two waves of amplitude $0.03\,\text{m}$ travelling in opposite directions, each with wavelength $3.0\,\text{m}$, frequency $60\,\text{Hz}$ and speed $180\,\text{m s}^{-1}$. (c) $648\,\text{N}$.
A stationary wave has separated position and time factors, as in (a), $2\cos3x\sin10t$. Expression (b) is a function of $x-vt$, so it is a travelling wave where the square root is defined. In (c), both terms depend on the same travelling combination $5x-0.5t$, so their sum can be rewritten as a single travelling wave of that argument. Expression (d) is a sum of two stationary-wave terms with different angular frequencies and wave numbers; it is not a single travelling wave and has no fixed standing-wave pattern.
(a) stationary wave; (b) travelling wave; (c) travelling wave; (d) none of these as a single travelling or stationary wave.
The wire length is $L=m/\mu=(3.5\times10^{-2})/(4.0\times10^{-2})=0.875\,\text{m}$. In the fundamental mode of a string fixed at both ends, $f=v/(2L)$, so $v=2Lf=2(0.875)(45)=78.75\,\text{m s}^{-1}$. The tension is $T=\mu v^2=4.0\times10^{-2}(78.75)^2=2.48\times10^2\,\text{N}$.
(a) $78.8\,\text{m s}^{-1}$; (b) $2.48\times10^2\,\text{N}$.
For a tube closed at one end and open at the other, consecutive resonance lengths differ by $\lambda/2$. Thus $\lambda/2=79.3\,\text{cm}-25.5\,\text{cm}=53.8\,\text{cm}=0.538\,\text{m}$, so $\lambda=1.076\,\text{m}$. With $f=340\,\text{Hz}$, $v=f\lambda=340(1.076)=3.66\times10^2\,\text{m s}^{-1}$.
$3.66\times10^2\,\text{m s}^{-1}$.
The clamped middle is a displacement node and the two free ends are antinodes. In the fundamental mode, each half of the rod is one quarter wavelength, so the full rod length is $L=\lambda/2$. Thus $\lambda=2L=2.00\,\text{m}$. With $f=2.53\,\text{kHz}=2.53\times10^3\,\text{Hz}$, $v=f\lambda=2.53\times10^3\times2.00=5.06\times10^3\,\text{m s}^{-1}$.
$5.06\times10^3\,\text{m s}^{-1}$.
For a pipe closed at one end, allowed frequencies are $(2n-1)v/(4L)$. With $L=0.20\,\text{m}$, the fundamental is $v/(4L)=340/0.80=425\,\text{Hz}$, close to $430\,\text{Hz}$, so the fundamental mode is resonantly excited. For a pipe open at both ends, the allowed frequencies are $nv/(2L)=n(340/0.40)=850n\,\text{Hz}$. Since $430\,\text{Hz}$ is not an allowed open-pipe frequency, resonance will not occur.
For the closed pipe, the source excites the fundamental mode approximately. If both ends are open, the source is not in resonance.
Reducing the tension in string A reduces its frequency. Since the beat frequency also decreases, string A must originally have had a frequency greater than string B. Therefore $f_A-f_B=6\,\text{Hz}$. With $f_A=324\,\text{Hz}$, $f_B=324-6=318\,\text{Hz}$.
$318\,\text{Hz}$.
(a) In a sound standing wave, pressure variation is related to compression and rarefaction, which are maximum where neighbouring particles move oppositely around a displacement node. At a displacement antinode, neighbouring particles move together and pressure variation is minimum. (b) Bats emit ultrasonic pulses and analyse the reflected echoes; time delay gives distance, direction comes from the receiving geometry, and echo strength and structure give information about size and nature. (c) A violin and sitar may have the same fundamental frequency, but their harmonic content, relative intensities and transients are different, so their quality or timbre differs. (d) Transverse mechanical waves require shear restoring forces. Solids have shear modulus and can support both transverse and longitudinal waves, whereas gases have essentially no shear modulus and support only longitudinal pressure waves. (e) A pulse is a superposition of many frequency components. In a dispersive medium, these components travel with different speeds, so their relative phases change and the pulse shape is distorted.
(a) Pressure variation is greatest where displacement is zero and least where displacement is greatest. (b) Bats use echoes of ultrasonic waves. (c) The notes differ in waveform and overtones. (d) Solids have shear elasticity; gases do not. (e) Different frequency components travel at different speeds in a dispersive medium.