Waves

For a guitar string, the wavelength of the -th harmonic is given by: where: is the length of the string, and is the harmonic number. Substituting the values:

In shallow water, the wave speed decreases, causing the waves to change direction and diffract. This is due to the change in the properties of the medium when the waves move from deeper to shallow regions.

The de-Broglie wavelength of a thermal neutron is given by:

The frequency at which an observer hears sound from a moving source is affected by the Doppler effect.
The observed frequency increases when the observer is moving toward the source or when the source is moving toward the observer.
In this case, if the observer moves toward the source while the source is moving away from the observer more slowly, the frequency will increase.
Thus, the correct answer is (d).

The frequency of a vibrating string is determined by the length of the string, the tension, and the mass per unit length.
The fundamental frequency is given by the equation: where is the length of the string, and is the wave speed. When the string is plucked at a point 25 cm from one end, the effective length of the vibrating portion of the string is reduced, which increases the frequency.
Since the frequency is inversely proportional to the length of the vibrating portion, the frequency is doubled.
Thus, the correct answer is (c).

The velocity of sound in a gas is given by the equation: where is the velocity of sound, is the pressure, is the adiabatic index, and is the density of the gas. When the pressure is increased by a factor of 4, without changing the temperature, the density also increases in proportion to the pressure. However, since the temperature does not change, the velocity of sound remains the same. Therefore, the velocity of sound will still be 330 ms .

The displacement for a simple harmonic wave is given by:

where is the amplitude, is the wave number, is the position, and is the angular frequency. The displacement depends on both the position and the wave's wavelength , since the wave number is related to the wavelength by the equation .

Using the given information, we can find the wavelength by solving for using the relationship between the wave number and the wavelength . Once we have the correct values and solve for , we find that the wavelength is 250 cm.

Thus, the correct answer is (d).

We can use the Doppler Effect formula to estimate the frequencies. If we know the difference in frequencies and the speed of sound, we can find the other frequency. Given that the difference is 4 Hz and the speed of sound is 340 m/s, solving for the other frequency gives: where and . Thus, the second frequency is: Hence, the correct answer is (d).