Question

Why does doubling the amplitude of a sound wave quadruple its intensity while halving its wavelength only halves its intensity?

Answer 1

The relationship between amplitude, wavelength, and intensity in a wave is governed by the physics of wave propagation. Let's explore the concepts involved to understand why doubling the amplitude quadruples the intensity, while halving the wavelength only halves the intensity.

1. Amplitude and Intensity: The amplitude of a wave represents the maximum displacement of particles in the medium from their equilibrium position. It determines the energy carried by the wave. The intensity of a wave, on the other hand, measures the energy transfer per unit area per unit time.

When a wave passes through a given area, the energy it carries is distributed over that area. The intensity is directly proportional to the square of the amplitude of the wave. Doubling the amplitude results in a fourfold increase in the amount of energy carried, which leads to a quadrupling of the intensity. This relationship arises from the conservation of energy and the distribution of that energy over an area.

Mathematically, the relationship between amplitude (A) and intensity (I) can be expressed as:

I ∝ A^2

2. Wavelength and Intensity: The wavelength of a wave represents the distance between two consecutive points in the wave that are in phase (e.g., two crests or two troughs). It determines the spatial extent of the wave. The intensity of a wave, however, depends on the energy transferred over a given area per unit time.

When the wavelength of a wave changes, the same amount of energy is spread over a different distance. Intensity is inversely proportional to the square of the wavelength. Therefore, halving the wavelength means that the energy is now spread over a smaller distance, resulting in a decrease in intensity by a factor of four.

Mathematically, the relationship between wavelength (λ) and intensity (I) can be expressed as:

I ∝ 1/λ^2

So, halving the wavelength reduces the intensity by a factor of 1/4 (or halves the intensity), whereas doubling the amplitude increases the intensity by a factor of 4 (or quadruples the intensity). These relationships hold true in various wave phenomena, including sound waves.