Waves

To determine the velocity with which an observer should approach a stationary sound source so that the apparent frequency becomes double the actual frequency, we can use the Doppler effect formula for sound. The formula for the apparent frequency ( ) perceived by an observer approaching a stationary sound source is given by:

Where:

• = actual frequency of the sound source
• = speed of sound in air
• = velocity of the observer

Since we want the apparent frequency to be double the actual frequency, we set up the equation:

Dividing both sides by , we get:

Multiplying both sides by results in:

Solving for v_o, we subtract from both sides:

Therefore, the observer should approach the stationary sound source with a velocity equal to the speed of sound ( ) to perceive the frequency as double the actual frequency.

The correct answer is:

In Young's double slit experiment, the intensity at any point on the screen is given by: where is the maximum intensity and is the phase difference between the two waves.
For bright and dark fringes, the intensity can be written in terms of the amplitudes and of the two waves as follows:

- For bright fringes:
- For dark fringes:
Given that the intensity ratio between bright and dark fringes is 3, we can equate the intensities: Substituting the expressions for and : Taking square roots on both sides: Solving for : Expanding: Rearranging terms: Thus:
Hence, the ratio of the amplitudes of the two waves is .
Thus, the correct answer is:

To solve the problem of finding the velocity of the car, we utilize the Doppler effect, which describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source.

The frequency of the sound heard by the driver after it reflects off the cliff can be related to the original frequency and the velocity of the car, , using the Doppler Effect formula for sound:

First, consider the sound wave emitted by the car horn as it approaches the cliff:

Where:

• is the emitted frequency.
• is the velocity of sound.
• is the velocity of the car.

Next, the sound wave reflects off the cliff and moves back towards the car. As the car is also moving towards the source (cliff), the frequency observed by the driver is altered again:

Given that the overall reflected frequency , we substitute for :

Simplifying this equation:

Cross-multiply to solve for :

Thus, the velocity of the car is:

Therefore, the correct answer is:

To calculate the minimum value of for constructive interference between successive pulses, we use the concept of the wave speed and the condition for constructive interference. The wave speed on a string is given by: Where:
- is the tension in the string (2.5 N),
- is the linear mass density of the string, which is given by , where is the mass of the string and is its length.
Substituting the values: - , - . The linear mass density is: Now, the wave speed is: The time interval for constructive interference is given by the time it takes for the wave pulse to travel a full wavelength. For constructive interference, this time corresponds to the period of the wave: Where is the wavelength of the wave. For constructive interference, the wavelength corresponds to twice the distance traveled by a wave pulse, which implies the pulse travels half the wavelength in the time . Given that the minimum value for constructive interference corresponds to a time of seconds, we find: Thus, the minimum value of is 20 ms.

The natural frequencies of a pipe closed at one end are given by the formula: Where:
- is the frequency of the -th harmonic,
- is the velocity of sound,
- is the length of the pipe. The frequencies for a closed-end pipe are given by the odd multiples of the fundamental frequency, so: We need to find the maximum such that . Calculating : Now we calculate the higher harmonics: Since the frequency exceeds 1250 Hz, we consider harmonics up to , which gives 6 possible oscillations (i.e., ).
Thus, the number of possible natural oscillations is .