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In wave physics, understanding wavelength is crucial as it defines the distance between consecutive points in phase on a wave, such as crest to crest or trough to trough. The wavelength is represented by the Greek letter \( \lambda \). To find the wavelength of a wave, use the equation:

• \( \lambda = \frac{v}{f} \) which connects the wave speed \( v \) and frequency \( f \).

Here, the speed of sound in air is given as \( 345 \text{ m/s} \) and the frequency is \( 524 \text{ Hz} \). Substituting these values in, you get \( \lambda = \frac{345}{524} \approx 0.658 \text{ meters} \). This means each cycle of the wave repeats approximately every 0.658 meters.

Wavelength can be affected by the medium through which the wave travels. For example, when moving from air to water, the wave speed changes, altering the wavelength accordingly while keeping the frequency constant. This behavior is key to understanding how sound and light waves propagate differently in various environments.

Frequency refers to how often the wave cycles, or repeats, occur in a second. It is measured in hertz (Hz), where one hertz is one cycle per second. Frequency is a defining property of waves, particularly sound and electromagnetic waves.

For the wave in question, the frequency is given as \( 524 \text{ Hz} \). This means that the wave undergoes 524 complete cycles every second. The link between frequency and wavelength is crucial: they are inversely proportional. That means, as frequency increases, the wavelength decreases if the wave speed remains constant.

It's also essential to understand the period of the wave, which is the time it takes for one complete cycle. The period \( T \) is the reciprocal of the frequency:

• \( T = \frac{1}{f} \)

For our example, \( T = \frac{1}{524} \approx 0.00191 \text{ seconds} \). Therefore, it takes roughly 0.00191 seconds for one cycle of the wave to pass a given point.

The phase of a wave refers to its position relative to a reference point and can be expressed in degrees or radians. Understanding phase difference helps in determining how "out of step" two waves or points on the same wave are compared.

• A full wave cycle corresponds to a phase change of \( 360^{\circ} \) or \( 2\pi \text{ radians} \).

For part (b) of the problem, calculating time for a \( 90^{\circ} \) phase change, which is a quarter of a complete cycle:

• The time required is \( \frac{T}{4} = \frac{0.00191}{4} \approx 0.0004775 \text{ seconds} \).

In part (c), when determining phase difference between two points 4.4 cm apart, interpret it using:

• \( \Delta \phi = \frac{360^{\circ}}{\lambda} \times d \)

With \( d = 0.044 \text{ m} \) and \( \lambda = 0.658 \text{ m} \), \( \Delta \phi \approx 24.05^{\circ} \). This reveals how in phase or out of phase two points are based on their separation.

Understanding phase difference is particularly important in technologies like noise-canceling headphones where opposing phases are used to nullify unwanted sound.