Why does the wave height of a tsunami increase as the tsunami enters shallow water?

• A. Because the $g$ravitational acceleration is stronger in shallow water, pushing the wave upward.
• B. Because the temperature of the water decreases in shallow areas, causing the wave to expand vertically.
• C. Because the $\rho$ density of water increases in shallow regions, forcing the wave to rise higher.
• D. Because the speed of the wave decreases, causing the $\lambda$wavelength to shorten and the energy to be concentrated in a smaller volume, which increases the wave height.

Answer: (D)

Verified step by step guidance

1. Understand that a tsunami is a type of wave that carries energy across the ocean, and its energy depends on both its speed and wavelength.
2. Recall the wave relationship: the wave speed \(v\) is related to the wavelength \(\lambda\) and frequency \(f\) by the equation \(v = \lambda \times f\).
3. As the tsunami approaches shallow water, the wave speed \(v\) decreases because the water depth affects the wave speed; shallower water means slower waves.
4. Since the frequency \(f\) of the wave remains constant (waves are generated by the same source), a decrease in speed \(v\) causes the wavelength \(\lambda\) to decrease (shorten).
5. Because the wave's energy is conserved but now compressed into a shorter wavelength, the energy density increases, causing the wave height (amplitude) to increase as the wave enters shallow water.