Question 21:
moderateSound waves travel at $350text{ m/s}$ through a warm air and at $3500text{ m/s}$ through brass. The wavelength of a $700text{ Hz}$ acoustic wave as it enters brass from warm air:
(2011 Pre)
Frequency $f$ remains constant when a wave changes medium. Velocity is given by $v = f\lambda$, which means $\lambda \propto v$. The ratio of velocities is $v_{brass}/v_{air} = 3500/350 = 10$. Thus, the wavelength increases by a factor of 10.
Question 22:
moderateA transverse wave is represented by $y = A\sin(\omega t – kx)$. For what value of the wavelength is the wave velocity equal to the maximum particle velocity?
(2010 Pre)
The maximum particle velocity is $v_{max} = A\omega$. The wave velocity is $v = \omega/k$. Equating the two gives $\omega/k = A\omega$, so $1/k = A$. Since $k = 2\pi/\lambda$, we get $\lambda/(2\pi) = A$, which yields $\lambda = 2\pi A$.
Question 23:
moderateA wave in a string has an amplitude of $2text{ cm}$. The wave travels in the +ve direction of x axis with a speed of $128text{ m/sec}$ and it is noted that 5 complete waves fit in $4text{ m}$ length of the string. The equation describing the wave is
(2009)
Amplitude $A = 2text{ cm} = 0.02text{ m}$. Wavelength $\lambda = 4/5 = 0.8text{ m}$. Wave number $k = 2\pi/0.8 \approx 7.85text{ m}^{-1}$. Angular frequency $\omega = vk = 128 \times 7.85 \approx 1005text{ rad/s}$. For +ve x-direction, $y = A\sin(kx - \omega t) = 0.02\sin(7.85x - 1005t)$.
Question 24:
moderateTwo points are located at a distance of $10text{ m}$ and $15text{ m}$ from the source of oscillation. The period of oscillation is $0.05text{ sec}$ and the velocity of the wave is $300text{ m/sec}$. What is the phase difference between the oscillations of two points?
(2008)
Frequency $f = 1/T = 1/0.05 = 20text{ Hz}$. Wavelength $\lambda = v/f = 300/20 = 15text{ m}$. The path difference is $\Delta x = 15 - 10 = 5text{ m}$. The phase difference is $\Delta\phi = (2\pi/\lambda)\Delta x = (2\pi/15) \times 5 = 2\pi/3$.
Question 25:
moderateA transverse wave propagating along x-axis is represented by $y(x,t) = 8.0\sin(0.5\pi x – 4\pi t – \pi/4)$ where $x$ is in metres and $t$ is in seconds. The speed of the wave is:
(2006)
From the given wave equation $y = A\sin(kx - \omega t - \phi)$, we identify $k = 0.5\pi\text{ m}^{-1}$ and $\omega = 4\pi\text{ rad/s}$. The wave speed is given by $v = \omega/k = 4\pi / 0.5\pi = 8\text{ m/s}$.
Question 26:
moderateThe temperature at which the speed of sound becomes double as was at $27^{\circ}\text{C}$ is:
(1993)
Since $v \propto \sqrt{T}$, to double the speed, temperature must be quadrupled. $T_1 = 273 + 27 = 300 \text{ K}$. $T_2 = 4 \times 300 = 1200 \text{ K}$, which is $1200 - 273 = 927^{\circ}\text{C}$.
Question 27:
moderateIf the amplitude of sound is doubled and the frequency reduced to one fourth, the intensity of sound at the same point will be:
(1989)
Intensity $I \propto A^2 f^2$. New intensity $I' \propto (2A)^2 (f/4)^2 = 4A^2 \times \frac{f^2}{16} = \frac{1}{4} I$. Thus it decreases by a factor of 4.
Question 28:
moderateThe velocity of sound in any gas depends upon:
(1988)
The velocity of sound in a gaseous medium is determined by the formula $v = \sqrt{\frac{E}{\rho}}$, showing dependence on elasticity ($E$) and density ($\rho$).
Question 29:
moderateEquation of progressive wave is given by $y = 4 \sin[\pi(\frac{t}{5} – \frac{x}{9}) + \frac{\pi}{6}]$ where $y, x$ are in cm and $t$ is in seconds. Then which of the following is correct?
(1988)
Comparing with $y = A \sin(2\pi(\frac{t}{T} - \frac{x}{\lambda}) + \phi)$, rewrite the given equation as $y = 4 \sin[2\pi(\frac{t}{10} - \frac{x}{18}) + \frac{\pi}{6}]$. This gives $\lambda = 18 \text{ cm}$.
Question 30:
moderateA wave travelling in positive $x$-direction with $A = 0.2 \text{ m}$ velocity $= 360 \text{ m/s}$ and $\lambda = 60 \text{ m}$, then correct expression for the wave is:
(2002)
Frequency $n = \frac{v}{\lambda} = \frac{360}{60} = 6 \text{ Hz}$. Equation for wave in positive x-direction is $y = A \sin[2\pi(nt - \frac{x}{\lambda})] = 0.2 \sin[2\pi(6t - \frac{x}{60})]$.