Two harmonic waves moving in the same direction superimpose to form a wave x = a cos (1.5t) cos (50.5t)$x = a \cos (1.5t) \cos (50.5t)$ where t$t$ is in seconds. Find the period with which they beat (close to nearest integer)
A.6 mathrm~s$6 \mathrm{~s}$
B.4 mathrm~s$4 \mathrm{~s}$
C.1 mathrm~s$1 \mathrm{~s}$
D.2 mathrm~s$2 \mathrm{~s}$
Solution & Explanation
### Related Formula
The product of cosines can be transformed into a sum using the trigonometric identity:
cos A cos B = frac12 [cos(A + B) + cos(A - B)]$$\cos A \cos B = \frac{1}{2} [\cos(A + B) + \cos(A - B)]$$
The beat frequency f_textbeat$f_{\text{beat}}$ is given by:
f_textbeat = |f_1 - f_2| = left| fracomega_1 - omega_22pi right|$$f_{\text{beat}} = |f_1 - f_2| = \left| \frac{\omega_1 - \omega_2}{2\pi} \right|$$
The beat period T_textbeat$T_{\text{beat}}$ is:
T_textbeat = frac1f_textbeat$$T_{\text{beat}} = \frac{1}{f_{\text{beat}}}$$
### Core Logic
Rewrite the superposition equation:
x = a cos(1.5t) cos(50.5t)$$x = a \cos(1.5t) \cos(50.5t)$$
Apply the identity with A = 50.5t$A = 50.5t$ and B = 1.5t$B = 1.5t$:
x = fraca2 [cos(52t) + cos(49t)]$$x = \frac{a}{2} [\cos(52t) + \cos(49t)]$$
Here, the two component frequencies are:
omega_1 = 52 mathrm~rad/s implies f_1 = frac522pi$$\omega_1 = 52 \mathrm{~rad/s} \implies f_1 = \frac{52}{2\pi}$$omega_2 = 49 mathrm~rad/s implies f_2 = frac492pi$$\omega_2 = 49 \mathrm{~rad/s} \implies f_2 = \frac{49}{2\pi}$$
Calculate the beat frequency:
f_textbeat = f_1 - f_2 = frac52 - 492pi = frac32pi mathrm~Hz$$f_{\text{beat}} = f_1 - f_2 = \frac{52 - 49}{2\pi} = \frac{3}{2\pi} \mathrm{~Hz}$$
### Step 1: Calculate Beat Period
The time period of beats is:
T_textbeat = frac1f_textbeat = frac2pi3 approx frac2 times 3.143 = 2.09 mathrm~s$$T_{\text{beat}} = \frac{1}{f_{\text{beat}}} = \frac{2\pi}{3} \approx \frac{2 \times 3.14}{3} = 2.09 \mathrm{~s}$$
Rounding to the nearest integer gives 2 mathrm~s$2 \mathrm{~s}$.
### Pattern Recognition
Sees: product of two cosines with significantly different coefficients omega_1$\omega_1$ and omega_2$\omega_2$.
Shortcut: The beat period is simply 2pi$2\pi$ divided by the difference between the two component frequencies, where the component frequencies are (omega_textaverage pm omega_textenvelope)$(\omega_{\text{average}} \pm \omega_{\text{envelope}})$. The difference is 2 times omega_textenvelope = 2 times 1.5 = 3 mathrm~rad/s$2 \times \omega_{\text{envelope}} = 2 \times 1.5 = 3 \mathrm{~rad/s}$. Thus, T = 2pi / 3 approx 2 mathrm~s$T = 2\pi / 3 \approx 2 \mathrm{~s}$.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Physics: Waves
Keywords:#period with which they beat#JEE Main 2025 Morning Q1#Waves JEE Main 2025#Beats and wave superposition
More Waves Previous-Year Questions — Page 4
Q45jee_main_2024_31_jan_morningOrgan Pipes
The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is 60mathrm~cm$60\mathrm{~cm}$, the length of the closed pipe will be:
A.60mathrm~cm$60\mathrm{~cm}$
B.45mathrm~cm$45\mathrm{~cm}$
C.30mathrm~cm$30\mathrm{~cm}$
D.15mathrm~cm$15\mathrm{~cm}$
Solution
### Related Formula
f_textclosed, fundamental = fracv4L_c$$f_{\text{closed, fundamental}} = \frac{v}{4L_c}$$f_textopen, 1st overtone = frac2v2L_o$$f_{\text{open, 1st overtone}} = \frac{2v}{2L_o}$$
### Core Logic
Organ Pipes diagram for Q45 - JEE Main 2024 MorningOrgan Pipes diagram for Q45 - JEE Main 2024 Morning
For a closed organ pipe, the fundamental frequency (1st harmonic) is:
f_1 = fracvlambda = fracv4L_1$$f_1 = \frac{v}{\lambda} = \frac{v}{4L_1}$$
where L_1$L_1$ is the length of the closed pipe.
For an open organ pipe, the first overtone (2nd harmonic) is:
f_2 = frac2v2L_2 = fracvL_2$$f_2 = \frac{2v}{2L_2} = \frac{v}{L_2}$$
where L_2$L_2$ is the length of the open pipe (L_2 = 60mathrm\,cm$L_2 = 60\mathrm{\,cm}$).
### Step 2: Equating Frequencies
Given f_1 = f_2$f_1 = f_2$:
fracv4L_1 = fracvL_2$$\frac{v}{4L_1} = \frac{v}{L_2}$$L_2 = 4L_1$L_2 = 4L_1$60 = 4 times L_1$$60 = 4 \times L_1$$L_1 = 15mathrm\,cm$$L_1 = 15\mathrm{\,cm}$$
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Physics: Waves
More Waves Questions — jee_main_2025_07_april_morning
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