Waves

List-I shows four configurations made of straight and semi-circular narrow tubes containing air. A sound wave of wavelength $\lambda=0.29$ m enters these structures at the point $S$ and a sound detector is placed at $D$. Between the points $S$ and $D$, the sound travels only through the tubes. List-II contains the possible smallest values of $l$ (refer to the figures) for which the detector $D$ records maximum amplitude. Ignore effects of sharp corners. [Given $\cos(15^\circ)=0.97$]

Choose the option that best describes the match between the entries in List-I to those in List-II.

The equation of a plane progressive wave is given by $y=5\cos\pi\left(200t-\dfrac x{150}\right)$ where $x$ and $y$ are in cm and $t$ is in second. The velocity of the wave is ________ m/s.

A transverse wave on a string is described by $y=3\sin(36t+0.018x+\pi/4)$, where $x$, $y$ are in cm and $t$ in seconds. The least distance between the two successive crests in the wave is ________ cm. (Nearest integer)
($\pi=3.14$)

Consider a system of three connected strings, $S_1$, $S_2$ and $S_3$ with uniform linear mass densities $\mu$ kg/m, $4\mu$ kg/m and $16\mu$ kg/m, respectively, as shown in the figure. $S_1$ and $S_2$ are connected at the point $P$, whereas $S_2$ and $S_3$ are connected at the point $Q$, and the other end of $S_3$ is connected to a wall. A wave generator O is connected to the free end of $S_1$. The wave from the generator is represented by $y = y_0\cos(\omega t - kx)$ cm, where $y_0$, $\omega$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct:

An audio transmitter (T) and a receiver (R) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the $X$ axis. They are pulled from the equilibrium position in opposite directions along the $X$ axis by a small angular amplitude $\theta_0 = \cos^{-1}(0.9)$ and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is 330 m/s, the maximum variation in the frequency (in Hz) as measured by the receiver (Take the acceleration due to gravity $g = 10\ \mathrm{m/s^2}$) is _____.

Two uniform strings of mass per unit length $\mu$ and $4\mu$, and length $L$ and $2L$, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $\nu_0 = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$, which of the following statement(s) is(are) correct?

A source (S) of sound has frequency $240\,\mathrm{Hz}$. When the observer (O) and the source move towards each other at a speed $v$ with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be $288\,\mathrm{Hz}$. However, when the observer and the source move away from each other at the same speed $v$ with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be $n\,\mathrm{Hz}$. The value of $n$ is _____.

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $\mathrm v_n$. At a fixed temperature and pressure, which of the following is the correct option?

A string of length 1 m and mass $2\times10^{-5}$ kg is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies 750 Hz and 1000 Hz. The value of tension $T$ is _____ Newton.

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in Hz is _____.

Consider both sources emitting sound. When $S_2$ is at $R$ and $S_1$ approaches the detector with a speed $4\ \mathrm{m\,s^{-1}}$, the beat frequency measured by the detector is _____ Hz.

The equations of two waves are given by :
$$y_1=5\sin2\pi(x-vt)\ \text{cm}$$
$$y_2=3\sin2\pi(x-vt+1.5)\ \text{cm}$$
These waves are simultaneously passing through a string. The amplitude of the resulting wave is :

Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by
$$y=\left(10\cos\pi x\sin\frac{2\pi t}T\right)\text{cm}$$
The amplitude of the particle at $x=\dfrac43$ cm will be ______ cm.

The first overtone frequency of an open organ pipe is equal to the fundamental frequency of a closed organ pipe. If the length of the closed organ pipe is 20 cm. The length of the open organ pipe is ______ cm.

A set of 20 tuning forks is arranged in a series of increasing frequencies. If each fork gives 4 beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is ______ Hz.

An observer moves towards a stationary source of sound with a velocity equal to one-fifth of the velocity of sound. The percentage change in the frequency will be :

If a wave gets refracted into a denser medium, then which of the following is true?

The velocity of sound in a gas, in which two wavelengths 4.08m and 4.16m produce 40 beats in 12s, will be :

A tunning fork of frequency 340 Hz resonates in the fundamental mode with an air column of length 125 cm in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is ______ cm.

(Velocity of sound in air is 340 ms$^{-1}$)

A longitudinal wave is represented by $x=10\sin2\pi\left(nt-\dfrac x\lambda\right)$ cm. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to :