Waves – NEET UG PhysicsPractice Questions & PYQs

Comparing the phase difference between two wave functions requires that both equations be expressed using the same trigonometric operator. The first wave function is $y_{1}=10^{-6} \sin \left[100 t+\left(x / 50\right)+0.5\right]$, which establishes its phase as $100 t+\left(x / 50\right)+0.5$. To transform the second wave $y_{2}=10^{-6} \cos \left[100 t+\left(x / 50\right)\right]$ into a sine format, we apply the trigonometric identity $\cos(\theta) = \sin(\theta + \pi/2)$. This substitution results in $y_{2}=10^{-6} \sin \left[100 t+\left(x / 50\right)+\pi/2\right]$, where the constant phase term $\pi/2$ is approximately 1.57. Subtracting the phase of the first wave from the phase of the second, we perform the calculation 1.57 minus 0.5, which yields a total phase difference of 1.07 radians.

The Doppler effect describes how the perceived frequency of a sound wave shifts due to relative motion, and when an observer moves toward a stationary source, they intercept wave fronts more frequently. The apparent frequency is calculated using the relationship $f' = \frac{v + v_o}{v} f$, where substituting the observer's speed of $v_o = \frac{v}{5}$ yields $f' = \frac{v + 0.2v}{v} f = 1.2 f$. Unlike frequency, the wavelength is a property of the wave determined by the source and the medium; because the source is stationary relative to the air, it continues to emit waves with the original spacing, meaning the wavelength remains unchanged at $\lambda$.

A wave traveling in the positive $x$-direction is described by the general equation $y = A \sin [2 \pi (f t - x / \lambda)]$, where the negative sign within the argument reflects the direction of propagation. The frequency $f$ is found by dividing the wave velocity $v = 360 \text{ m/s}$ by the wavelength $\lambda = 60 \text{ m}$, which yields $f = 360 / 60 = 6 \text{ Hz}$. By substituting the amplitude $A = 0.2 \text{ m}$, the calculated frequency of $6 \text{ Hz}$, and the given wavelength of $60 \text{ m}$ into the standard wave formula, the resulting expression is $y = 0.2 \sin [2 \pi (6 t - x / 60)]$.

The Doppler effect dictates that the pitch perceived by a listener varies based on the relative motion of the sound source along the line of sight. For a whistle rotating in a circle, the source moves tangent to its path at every instant. An observer positioned far from the center perceives the lowest possible frequency when the whistle is moving directly away from them, as this motion maximizes the expansion of the sound waves reaching the observer.

The linear velocity of the source is determined by the radius of $0.5 \text{ m}$ and the angular speed of $20 \text{ rad/s}$:
$v = r\omega = 0.5 \times 20 = 10 \text{ m/s}$
When the source recedes, the frequency $f'$ heard by the observer is calculated using the Doppler formula for a moving source:
$f' = f_0 \left( \frac{v_s}{v_s + v} \right)$
Substituting the actual frequency $f_0 = 385 \text{ Hz}$, the speed of sound $v_s = 340 \text{ m/s}$, and the source velocity $v = 10 \text{ m/s}$ results in:
$f' = 385 \left( \frac{340}{340 + 10} \right) = 385 \times \frac{340}{350} = 374 \text{ Hz}$

Superposition of two sinusoidal waves with identical frequencies and equal amplitudes results in a combined wave where the resultant amplitude depends on the phase difference between the sources. The square of the resultant amplitude $R$ is given by the relation
$R^2 = a^2 + a^2 + 2 a^2 \cos \phi$
where $a$ is the amplitude of the individual waves and $\phi$ is the phase difference. Given that the resultant amplitude is equal to the individual amplitude $a$, we substitute $R = a$ into the equation to obtain $a^2 = 2 a^2 + 2 a^2 \cos \phi$. Dividing both sides by $2 a^2$ allows us to isolate the phase term as $1 + \cos \phi = \frac{1}{2}$. Consequently, $\cos \phi = -\frac{1}{2}$, which identifies the phase difference as $\frac{2 \pi}{3}$.

The general form of a traveling wave equation is $y = A \sin(\omega t - kx)$, where $\omega$ represents the angular frequency associated with time and $k$ is the wave number associated with spatial displacement. By identifying the terms in the given equation $y = 10^{-4} \sin(100t - \frac{x}{10})$, we find that $\omega = 100$ and $k = 1/10$. The velocity of the wave is calculated using the relationship $v = \omega / k$, and substituting these values results in $v = 100 / (1/10)$, which equals $1000$ m/s.

The fundamental frequency of a sonometer wire follows the relationship $n = \frac{1}{2l} \sqrt{\frac{T}{\pi r^2 \rho}}$, where $T$ represents the tension, $r$ the radius, and $\rho$ the material density. Adjusting the parameters by doubling the tension to $2T$, increasing the radius to $2r$ to account for the doubled diameter, and halving the density to $\frac{\rho}{2}$ results in the new frequency $n' = \frac{1}{2l} \sqrt{\frac{2T}{\pi (2r)^2 (\frac{\rho}{2})}}$. Simplifying the denominator yields $\pi (4r^2) (\frac{\rho}{2})$, which reduces to $2 \pi r^2 \rho$; because the factor of $2$ in the numerator and the denominator cancels out, the expression returns to its original form, confirming that the new frequency $n'$ is identical to the original frequency $n$.

The standard expression for a sinusoidal wave is $y = A \cos(\omega t + \phi)$, where $\omega$ represents the angular frequency. By comparing this to the given wave equation $y = 0.40 \cos(2000 t + 0.80)$, it is evident that $\omega$ equals 2000. The relationship between angular frequency and linear frequency $\nu$ is defined by the following equation:

$2 \pi \nu = 2000$

Isolating $\nu$ by dividing by $2 \pi$ leads to a frequency of $\nu = \frac{1000}{\pi}$ Hz.

The speed at which sound travels through a gas is determined by the inherent physical properties of the medium itself rather than the characteristics of the wave, such as its frequency or amplitude. Sound propagation relies on the gas's elasticity, which acts as a restoring force, and its density, which determines the inertia of the particles. This fundamental relationship is mathematically expressed by the formula $v = \sqrt{\frac{B}{\rho}}$, which can also be written as $v = \sqrt{\frac{\gamma P}{\rho}}$ to account for the gas pressure and adiabatic index.

When two simple harmonic motions with identical amplitudes and frequencies oscillate along perpendicular axes, their combined motion traces a path determined by the phase difference between them. Given $x = a \cos(\omega t + \delta)$ and $y = a \cos(\omega t + \alpha)$ with the condition $\delta = \alpha + \frac{\pi}{2}$, substituting the value of $\delta$ into the equation for $x$ allows the use of the trigonometric identity $\cos(\theta + \frac{\pi}{2}) = -\sin(\theta)$. This simplifies the expression for $x$ to $x = -a \sin(\omega t + \alpha)$.

Combining these two expressions by calculating the sum of their squares yields $x^2 + y^2 = (-a \sin(\omega t + \alpha))^2 + (a \cos(\omega t + \alpha))^2$. Simplifying this expression using the trigonometric identity $\sin^2\theta + \cos^2\theta = 1$ results in $x^2 + y^2 = a^2$. This final equation describes a circle with radius $a$ centered at the origin, confirming that the resultant path of the two waves is circular.

The fundamental frequency $n$ of a stretched string is inversely proportional to its length $l$, described by the relationship $n \propto \frac{1}{l}$. Since the total original length $l$ is the sum of the three individual segment lengths $l_1$, $l_2$, and $l_3$, we can write $l = l_1 + l_2 + l_3$. Substituting the inverse frequency relationship into this length equation demonstrates that the reciprocal of the original frequency is equal to the sum of the reciprocals of the fundamental frequencies of the three segments:
$\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}$

When a sound source and an observer move toward each other, the observed frequency increases because sound waves become compressed in the space between them. We calculate this shift using the Doppler effect formula
$f' = f \left( \frac{v + u}{v - u} \right)$
where $f'$ is the observed frequency, $f$ is the original frequency, $v$ is the speed of sound, and $u$ is the speed of each train.

Given that the frequency changes to 9/8 of the original, the relationship becomes
$\frac{9}{8} = \frac{340 + u}{340 - u}$
Multiplying both sides leads to the equation $9(340 - u) = 8(340 + u)$, which simplifies to $3060 - 9u = 2720 + 8u$. Combining the terms gives $17u = 340$, which confirms that the speed of each train is 20 m/s.

The speed of sound in a gas is directly proportional to the square root of its absolute temperature, described by the relationship $v \propto \sqrt{T}$. To apply this proportionality, one must always convert temperatures to the Kelvin scale, meaning the initial temperature of $27^{\circ}\text{C}$ becomes $300\text{ K}$. When comparing two states where the velocity doubles, the ratio of velocities and temperatures is $\frac{v}{2v} = \sqrt{\frac{300}{T}}$, where $T$ is the unknown absolute temperature. Squaring both sides of this equation yields $\frac{1}{4} = \frac{300}{T}$, which calculates to $T = 1200\text{ K}$, equivalent to $927^{\circ}\text{C}$ on the Celsius scale.

The speed of a transverse wave travelling along a stretched string depends directly on the tension force applied to it and the material's linear mass density. This relationship is defined by the following expression:

$v = \sqrt{\frac{T}{\mu}}$

where $T$ represents the tension and $\mu$ is the mass per unit length. Finding the linear mass density first involves dividing the total mass by the string length, which gives $\mu = \frac{0.035}{5.5}$ kg/m. When you substitute the given tension of 77 N and this density value into the wave equation, the expression becomes $v = \sqrt{\frac{77}{0.035/5.5}}$. Simplifying this calculation results in $v = \sqrt{\frac{77 \times 5.5}{0.035}}$, which leads to a final wave speed of 110 m/s.

The phenomenon of beats occurs here due to the Doppler effect, where the observer moving at velocity $u$ perceives shifted frequencies from two stationary sources emitting waves of wavelength $\lambda$. As the observer approaches one source, the detected frequency becomes $\frac{v+u}{\lambda}$, while moving away from the second source results in a detected frequency of $\frac{v-u}{\lambda}$, based on the relationship between wave speed $v$ and wavelength $\lambda$. Since the beat frequency corresponds to the absolute difference between these two signals, the number of beats is determined by:

$\frac{v+u}{\lambda} - \frac{v-u}{\lambda} = \frac{2 u}{\lambda}$

This derivation confirms that the rate of beats heard by the observer is directly proportional to their velocity and inversely proportional to the wavelength.

The fundamental frequency of a vibrating stretched string is inversely proportional to its length when the tension and linear mass density remain constant. This relationship is defined by the equation $f = \frac{1}{2l} \sqrt{\frac{T}{\mu}}$, which implies that the product of frequency and length must remain constant, as expressed by $f_1 l_1 = f_2 l_2$. When the frequency is decreased from 512 Hz to 256 Hz, the frequency is effectively halved. To maintain this constant product, the length of the string must be doubled, which, starting from an initial length of 0.5 m, results in a final length of 1 m.