Question

Wave Steepness The steepness of a wave is represented by the formula Steepness \(=H / L\), where \(H=\) wave height and \(L=\) wavelength. When the steepness of a wave reaches \(1 / 7\), the wave becomes unstable and breaks. If \(L=50 \mathrm{~m}\), at what height will the wave break?

Short answer

The wave will break when its height is approximately 7.14 meters.

Step-by-step solution

Understand the Formula

The steepness of a wave is calculated by the formula \(\text{Steepness} = \frac{H}{L}\) where \(H\) is the wave height and \(L\) is the wavelength.

Substitute Known Values into the Formula

We know that the wave becomes unstable when its steepness reaches \(\frac{1}{7}\), and we are given that \(L = 50\, \text{m}\). Substitute these values into the formula: \[\frac{H}{50} = \frac{1}{7}\].

Solve for Wave Height \(H\)

To find \(H\), multiply both sides of the equation \(\frac{H}{50} = \frac{1}{7}\) by \(50\) to solve for \(H\): \[H = \frac{1}{7} \times 50\].

Calculate Wave Height

Carry out the multiplication to find \(H\): \[H = \frac{50}{7} \approx 7.14\, \text{m}\].

Key concepts

Wave Height

Wave height is a crucial factor in determining the steepness and stability of ocean waves. It is defined as the vertical distance between the crest (the highest point of the wave) and the trough (the lowest point). In the formula for wave steepness, \( H \) represents the wave height.

• A higher wave height means a steeper wave, which can lead to instability.
• The average wave height helps in predicting the energy and potential danger of waves, especially during storms or in rough seas.

Understanding wave height is essential, as it impacts everything from ship navigation to surfers catching the perfect wave. In this particular exercise, we calculated the wave height at which a wave becomes unstable, given that the wavelength is \( 50 \mathrm{~m} \). This height was approximately \( 7.14 \mathrm{~m} \).

Wavelength

Wavelength is the horizontal distance between two consecutive crests or troughs of a wave. In the context of wave steepness, \( L \) represents the wavelength. It plays a significant role in defining wave motion and behavior.

• Longer wavelengths generally indicate less steep waves.
• Shorter wavelengths can lead to steeper, more unstable waves.

In our exercise, the given wavelength was \( 50 \mathrm{~m} \), which, when used alongside the critical steepness ratio of \( 1/7 \), helped us find the wave height at which instability occurs. By knowing the wavelength, you can better predict how a wave will grow or dissipate over time.

Unstable Waves

Unstable waves are those that have reached a critical steepness level, making them prone to breaking. The critical steepness value is \( 1/7 \), meaning that once \[ \text{Steepness} = \frac{H}{L} = \frac{1}{7} \], the wave becomes unstable and breaks.

• Instability occurs when the wave height to wavelength ratio surpasses \( 1/7 \).
• Wave breaking due to instability can either be a spilling breaker, a plunging breaker, or a surging breaker, depending on the wave energy and seabed slope.

In the exercise, using the \( 1/7 \) ratio with a \( 50 \mathrm{~m} \) wavelength determined the wave height at which a wave becomes unstable is roughly \( 7.14 \mathrm{~m} \). Understanding the conditions leading to unstable waves helps in coastal management, minimizing erosion, and ensuring safety in marine activities.