Question

Wave Speed Wave speed of deep water waves is calculated using the formula \(S=L / T\), where \(S\) represents wave speed, \(L\) represents the wavelength, and \(T\) represents the period of the wave. What is the speed of a wave if \(L=100 \mathrm{~m}\) and \(T=11 \mathrm{~s}\) ?

Short answer

The speed of the wave is 9.09 m/s.

Step-by-step solution

Understand the Formula

The problem gives us the formula for wave speed: \( S = \frac{L}{T} \), where \( S \) is the wave speed, \( L \) is the wavelength, and \( T \) is the period.

Substitute Values into Formula

Substitute the given values into the formula: \( L = 100 \text{ m} \) and \( T = 11 \text{ s} \). Therefore, the formula becomes \( S = \frac{100}{11} \).

Perform the Calculation

Calculate the division: \( \frac{100}{11} = 9.09 \text{ m/s} \) (rounded to two decimal places).

Conclusion

The calculated wave speed \( S \) is \( 9.09 \text{ m/s} \).

Key concepts

Wavelength

Wavelength, often denoted as \(L\), is a fundamental property of all waves, including deep water waves. It is defined as the distance between two consecutive points that are in phase, such as from crest to crest or trough to trough in a wave. Understanding wavelength is essential because it helps to determine other wave properties like speed and frequency.

In our example, the wavelength was given as 100 meters. This tells us the length of one complete wave cycle from start to finish. This measurement is crucial for calculating the wave speed using the provided formula \(S = \frac{L}{T}\). The longer the wavelength, the more energy the wave can potentially carry.

Wave Period

The wave period, represented by \(T\), is the time it takes for a single wave to pass a fixed point. Think of it as the rhythm or pulse of the wave. It's measured in seconds and plays a key role in determining wave speed.

For our calculation, we have a wave period of 11 seconds. This means each wave completes its cycle and passes the same point every 11 seconds. Knowing the period is important because it allows us to use the wave speed formula \(S = \frac{L}{T}\) accurately. In our example, the period works in harmony with the wavelength to give us the correct wave speed.

• The longer the period, the slower the wave travels for a given wavelength.
• If the period decreases, waves travel faster assuming the same wavelength.

Deep Water Waves

Deep water waves are waves that travel in water where the depth is greater than half their wavelength. They behave differently from shallow water waves, as their speed depends primarily on the wavelength and period rather than the depth of the water.

In the context of our exercise, we calculate the wave speed in deep water, which validates using the simple formula \(S = \frac{L}{T}\). Because the depth is irrelevant in deep water wave calculations, the given formula stays valid whenever the conditions for deep water waves are met.

Key characteristics of deep water waves:

• They are not influenced by the sea floor, unlike shallow water waves.
• Their motion consists of circular orbits, diminishing with depth.

Understanding these properties allows us to predict wave behavior more effectively in the open ocean, which is essential for navigation and coastal management.