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The Doppler effect is a phenomenon observed when there is relative motion between a sound source and an observer. It manifests as a change in frequency and pitch of the sound accordingly. In the context of sonar systems like the one on a ship used for detecting objects underwater, such as whales, the Doppler effect plays a crucial role. When the sonar's sound waves hit a moving object, the waves are reflected with a change in frequency dependent upon the relative speed and direction of the object. If the object is moving towards the source, the frequency increases, and when it moves away, the frequency decreases.

This phenomenon is not just a curiosity—it's essential for navigation and detection in marine contexts, giving information about the speed and direction of the object in relation to the ship. During the computation in the provided exercise, it is important to consider the sound source at rest relative to the water, ensuring that the observer's velocity in the Doppler effect formula is set to zero. The minus sign in the velocity of the source indicates that the whale is moving towards the ship, leading to an increase in the observed frequency.

Wavelength calculation is a fundamental aspect of physics that underlies the functioning of sonar systems. In the exercise, we apply the wave equation, which relates the speed of sound in a medium, the frequency of the wave, and its wavelength (denoted by \(\lambda\)). The formula \(\lambda = \frac{v}{f}\) allows us to calculate the distance between consecutive wave crests, which is essential for sonar operation since it determines the resolution and range of the sonar signal.

In our specific case, by dividing the speed of sound in water by the frequency at which the sonar system operates, we obtain the wavelength of the waves. The speed of sound in water is considerably different from in air, generally leading to longer wavelengths underwater. This impacts how well the sonar can detect and differentiate objects, making an accurate calculation of wavelength crucial for efficient sonar operation. In the exercise, you'll use the given constants, ensuring the resulting wavelength is in correct units, which is typically meters for such large-scale applications.

The speed of sound in water is an essential parameter for any calculation involving sonar systems. Unlike the speed of sound in air, which is about 343 meters per second (m/s) at room temperature, sound travels much faster in water, with an average speed of around 1482 m/s at \(20^\circ\mathrm{C}\). This speed can vary depending on various factors such as temperature, salinity, and depth of the water, which all affect how sonar systems are used and interpreted.

The speed of sound in water affects both the wavelength of the sonar waves and the interpretation of the Doppler effect. For instance, warmer water temperatures generally make sound waves travel faster. When solving physics problems involving sonar, it's crucial to use the correct speed value for accuracy. Notably, in the exercise provided, using the speed of sound in water allows us to calculate the wavelength and understand the effect of an object's velocity on the frequency perceived by the sonar system.