11. Sound isolation

Official Problem Statement

“The sound isolation of a room may become breached if the door is not shut properly, even if the gap is negligibly small. Investigate how the shape and size of a small opening affect the transmission of sound.”

Diffraction Through an Opening: Huygens’ Principle

When an opening is much smaller than the acoustic wavelength ($d ll lambda$), Huygens’ principle dictates that the opening acts as a point source, re-radiating sound in all directions into the half-space beyond the wall. Although the opening area is tiny, the transmitted power greatly exceeds what a simple area fraction would suggest, because the diffracted wave fills the entire half-space.

For a rigid wall with a circular opening of diameter $d$, the transmitted power in the long-wave regime ($kd ll 1$, where $k = 2pi/lambda$) scales as:

$$W_t approx W_i cdot tau,qquad tau approx frac{(kd)^4}{16}qquad (kd ll 1)$$

The transmission coefficient $tau$ increases strongly with frequency ($tau propto f^4$); at low frequencies the sub-wavelength opening is a very poor radiator, so even a tiny gap provides substantial isolation at bass frequencies.

Insertion Loss and Shape Dependence

The insertion loss (IL) is defined as:

$$text{IL} = 10log_{10}!left(frac{W_i}{W_t}right)quad (text{dB})$$

For a circular opening of diameter $d$ in the regime $kd ll 1$:

$$text{IL} approx -40log_{10}!left(frac{pi d}{lambda}right)$$

This shows that IL decreases with increasing frequency (at high $f$, $lambda$ is small, the opening is no longer in the deep sub-wavelength regime and becomes geometrically transparent).

Narrow slit versus circular hole: For a slit of width $d$ and length $L gg d$ with the same area $A = dL$, the radiation pattern is cylindrical rather than spherical. The far-field pressure pattern of the slit:

$$p(theta) propto frac{sin(pi dcostheta/lambda)}{pi dcostheta/lambda}$$

This pattern concentrates radiation along the slit axis. Compared to a circle of equal area, a long narrow slit generally has a lower IL (more leakage) because the effective radiation resistance differs for the two geometries.

General Scaling Rule

The two dominant parameters are the opening area $S$ and the wavelength $lambda$. In the limit $kd ll 1$, the transmission coefficient scales as:

$$tau sim left(frac{S}{lambda^2}right) cdot C_{text{shape}}$$

where $C_{text{shape}}$ is an order-unity factor depending on geometry — for a circle $C approx 0.12$, for a square $C approx 0.11$, and for a long slit $C$ is direction-dependent.

Parameters

Experimental Setup

To measure IL as a function of $d$ and frequency:

1. Mount a rigid partition with interchangeable openings (circles, squares, slits) between two coupled enclosures.
2. Drive a loudspeaker at discrete frequencies from 100 Hz to 5 kHz using a function generator.
3. Measure the sound pressure level (SPL) on both sides of the partition using calibrated microphones, with and without each opening.
4. Compute and plot $text{IL}(f,d)$ for each opening geometry.
5. To isolate shape effects, compare openings with equal area but different shapes.

Key practical finding: a 1 cm gap under a heavy door can reduce IL from 40 dB to less than 20 dB — a factor of 100 in transmitted power — demonstrating why even a small opening dominates the acoustic performance of a room boundary.

References

• Beranek, L. L., Acoustics, Acoustical Society of America, 1993 — Chapter on sound transmission through openings.
• Kuttruff, H., Room Acoustics, 5th ed., Spon Press, 2009 — Wall isolation with gaps.
• Pierce, A. D., Acoustics: An Introduction to Its Physical Principles and Applications, ASA, 2019.
• Rayleigh, Lord, Theory of Sound, Vol. 2, Macmillan, 1896 — Diffraction through small apertures.