Two waves of frequencies 20 Hz and 30 Hz. Travels out from a common point. The phase difference between them after 0.6 sec is

Options

A. zero
B. `(pi)/(2)`
C. `pi`
D. `(3 pi)/(4)`

Answer

To find the phase difference between two waves of frequencies 20 Hz and 30 Hz after 0.6 seconds, we can follow these steps:

### Step 1: Understand the relationship between frequency and angular frequency
The angular frequency (ω) is related to the frequency (f) by the formula:
\[ \omega = 2\pi f \]
where \(f\) is the frequency in Hertz (Hz).

### Step 2: Calculate the angular frequencies for both waves
For the first wave with a frequency of 20 Hz:
\[ \omega_1 = 2\pi \times 20 = 40\pi \, \text{rad/s} \]

For the second wave with a frequency of 30 Hz:
\[ \omega_2 = 2\pi \times 30 = 60\pi \, \text{rad/s} \]

### Step 3: Calculate the phase of each wave after 0.6 seconds
The phase (φ) of a wave at time \(t\) can be calculated using:
\[ \phi = \omega t \]

For the first wave:
\[ \phi_1 = \omega_1 \times t = 40\pi \times 0.6 = 24\pi \, \text{radians} \]

For the second wave:
\[ \phi_2 = \omega_2 \times t = 60\pi \times 0.6 = 36\pi \, \text{radians} \]

### Step 4: Calculate the phase difference
The phase difference (Δφ) between the two waves is given by:
\[ \Delta \phi = \phi_2 - \phi_1 \]
Substituting the values we calculated:
\[ \Delta \phi = 36\pi - 24\pi = 12\pi \, \text{radians} \]

### Step 5: Simplify the phase difference
Since phase differences are periodic with a period of \(2\pi\), we can reduce \(12\pi\) to find an equivalent phase difference:
\[ 12\pi \mod 2\pi = 0 \]
Thus, the phase difference can be expressed as:
\[ \Delta \phi = 0 \, \text{radians} \]

### Conclusion
The phase difference between the two waves after 0.6 seconds is:
\[ \Delta \phi = 0 \, \text{radians} \]