Wave Superposition Yields Beats
When two harmonic waves moving in the same direction superimpose, and their frequencies are close, the resultant wave exhibits a phenomenon called beating. This means the amplitude of the resultant wave varies periodically over time.
Component Wave Frequencies from Product Form
The given wave equation is $x = a \cos (1.5t) \cos (50.5t)$. This equation is a product of two cosine terms. To understand the superposition of two underlying harmonic waves, we use the trigonometric identity:
$ \cos A \cos B = \frac{1}{2} [\cos(A-B) + \cos(A+B)] $
Let $A = 50.5t$ and $B = 1.5t$. Applying the identity to the given equation:
$ x = \frac{a}{2} [\cos((50.5 - 1.5)t) + \cos((50.5 + 1.5)t)] $
$ x = \frac{a}{2} [\cos(49t) + \cos(52t)] $
This resulting form shows the superposition of two harmonic waves with angular frequencies $\omega_1 = 49 \text{ rad/s}$ and $\omega_2 = 52 \text{ rad/s}$.
Beat Period Calculation Steps
The difference between the angular frequencies of the two component waves determines the beating phenomenon. The difference is:
$ |\omega_2 - \omega_1| = |52 \text{ rad/s} - 49 \text{ rad/s}| = 3 \text{ rad/s} $
The beat frequency ($f_{beat}$) is half the difference between the angular frequencies, divided by $\pi$:
$ f_{beat} = \frac{|\omega_2 - \omega_1|}{2\pi} = \frac{3}{2\pi} \text{ Hz} $
The period of beating ($T_{beat}$) is the time it takes for one complete cycle of the amplitude variation. It is the reciprocal of the beat frequency:
$ T_{beat} = \frac{1}{f_{beat}} = \frac{2\pi}{|\omega_2 - \omega_1|} $
Substituting the calculated frequency difference:
$ T_{beat} = \frac{2\pi}{3} \text{ s} $
Result and Nearest Option
To find the value closest to an integer, we calculate the numerical value of $T_{beat}$:
$ T_{beat} \approx \frac{2 \times 3.14159}{3} \approx 2.094 \text{ s} $
Comparing this value to the given options, $2.094 \text{ s}$ is closest to $2 \text{ s}$.
Similar Questions
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- 5 beats/second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be
- A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x=0 at t=0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave ?
A particle is subjected to two simple harmonic motions as :
$x_1 = \sqrt{7} \sin 5t \text{ cm}$ and $x_2 = 2\sqrt{7} \sin \left(5t + \frac{\pi}{3}\right) \text{ cm}$
where x is displacement and t is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \text{ ms}^{-2}$. The value of x is :- In the resonance experiment, two air columns (closed at one end) of $100 \text{ cm}$ and $120 \text{ cm}$ long, give $15 \text{ beats per second}$ when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is:
A piston of mass $M$ is hung from a massless spring whose restoring force law goes as $F= -kx^3$, where $k$ is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with 'n' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature $T$) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height $L_0$ to $L_1$, the total energy delivered by the filament is : (Assume spring to be in its natural length before heating)
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
- The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2 = 50 - x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}~\text{s}$, the value of $x$ is ________.
Two simple harmonic motions, as shown below, are at right angles. They are combined to form lissajous figures.
$x(t) = A \sin (at + \delta)$
$y(t) = B \sin (bt)$
Identify the correct match below :
Important Questions from Oscillations and Waves
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- 5 beats/second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be
- A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x=0 at t=0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave ?
A particle is subjected to two simple harmonic motions as :
$x_1 = \sqrt{7} \sin 5t \text{ cm}$ and $x_2 = 2\sqrt{7} \sin \left(5t + \frac{\pi}{3}\right) \text{ cm}$
where x is displacement and t is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \text{ ms}^{-2}$. The value of x is :- In the resonance experiment, two air columns (closed at one end) of $100 \text{ cm}$ and $120 \text{ cm}$ long, give $15 \text{ beats per second}$ when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is:
