Understanding Sound Intensity, Wave Absorption, and Standing Waves

Sound Intensity and Loudness

The intensity of a sound wave, often referred to as volume, depends on the square of its frequency and the square of its amplitude. The human ear can detect sounds with an intensity starting from 10^-12 W m^-2, known as the threshold of hearing. At an intensity of 1 W m^-2, the pain threshold is reached. To quantify the sensation of loudness, we define the sound intensity level:

Where I is the sound intensity and I₀ is the threshold of hearing. The unit used to measure the level of loudness is the decibel (dB).

Wave Absorption and Intensity Decay

Beyond the decrease in intensity due to distance, waves also experience a reduction in energy caused by the medium through which they travel. This phenomenon, known as absorption, occurs because the medium is not perfectly elastic; friction between particles causes the wave to lose energy as it propagates. The decrease in intensity due to absorption is expressed by the following exponential decay function:

I = I₀ · e^-αx

Where:

• I: Intensity after traveling distance x.
• I₀: Initial intensity.
• α: Absorption coefficient of the medium (m^-1).

Deriving the Standing Wave Equation

To deduce the equation for a standing wave, we combine an incident wave and its reflection:

Incident wave: y₁(x, t) = A sin(ωt - kx)

Reflected wave: y₂(x, t) = A sin(ωt + kx)

The resulting wave Y = y₁ + y₂ is calculated as:

Y = A [sin(ωt)cos(kx) - cos(ωt)sin(kx) + sin(ωt)cos(kx) + cos(ωt)sin(kx)]

Y = 2A cos(kx) sin(ωt)

Are Standing Waves Real Waves?

A standing wave is not a traveling wave in the traditional sense. Instead, it is a superposition of simple harmonic motions in phase at all points, characterized by a variable amplitude modulated by a sinusoidal function.