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 _________ $\text{Hz}$.
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- 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 ________.
- 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 ________.
- 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
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 :
The motion of a mass on a spring, with spring constant K is as shown in figure.
The equation of motion is given by $x(t) = A\sin\omega t+B\cos\omega t$ with $\omega=\sqrt{\frac{K}{m}}$.
Suppose that at time $t = 0$, the position of mass is $x(0)$ and velocity $v(0)$, then its displacement can also be represented as $x(t) = C\cos(\omega t-\phi)$, where $C$ and $\phi$ are
- $T_0$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, the modified time period 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)
- 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 ?
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 _________ $\text{Hz}$.
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- 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 ________.
- The equation of a transverse wave is $y = y_0 \sin 2\pi(ft - \frac{x}{\lambda})$. If the maximum particle velocity be four times that of wave velocity then
- 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 ________.
