Inverse-square law

The inverse-square law says that sound intensity from an ideal point source spreads over a larger area as distance increases. In free-field conditions, intensity is proportional to 1/r2, where r is distance from the source. In practical sound-pressure terms, SPL drops by about 6 dB every time distance doubles.

The law is a useful first estimate for noise exposure around an industrial sonic horn, but it is not a complete model of cleaning performance inside a vessel.

Worked example for a sonic horn

A horn measured at 150 dB SPL at 1 m on the bell axis would be estimated as follows in a free field:

The calculation is usually written as level change equals 20 log10 of distance ratio for sound pressure, or 10 log10 for intensity. The familiar 6 dB rule is the rounded result for a doubling of distance.

Where the rule breaks down

Real industrial installations depart from the textbook model:

• Near field - close to the horn, pressure and particle velocity are not yet behaving as a simple radiating wave.
• Directivity - a bell horn sends more energy along its axis than behind or beside it.
• Reflections - vessel walls, tube banks and casing surfaces reflect sound and can raise levels above free-field estimates.
• Obstructions - baffles, bags, catalyst layers and duct turns create acoustic shadowing.
• Absorption - insulation, dust layers and high-frequency air absorption remove energy.
• Multiple sources - several horns firing together create a combined field, not a single-point field.

Why it matters for cleaning

Inside a boiler, ESP or hopper, acoustic cleaning often benefits from reflections. The target does not only receive direct sound from the bell; it also receives energy reflected by walls and internals. That is why horn placement is based on geometry, line of sight, wavelength and obstruction density rather than distance alone.

For open-area noise, however, the inverse-square law remains a useful screening tool. It helps estimate work-area levels, fence-line impact and the value of moving an operator station or access route farther from the horn.

Practical limits in plants

The inverse-square law assumes a point source in free space. Industrial vessels are rarely like that. A sonic horn mounted on a duct wall radiates through a bell, couples into a confined gas volume and reflects from steel, refractory, tube banks, catalyst modules and ash layers. Near the source, directivity and near-field behaviour dominate. Farther away, reflections and absorption can make the decay flatter or more irregular than a simple distance rule.

For worker-noise planning, the law is still a useful first screen. Doubling the distance from a small exposed source gives about a 6 dB reduction before reflections are considered. In a hard boiler house or between parallel casings, the actual reduction can be smaller because reflected sound adds to the direct path. Measurements should therefore be made at access platforms, walkways and normal work positions, not only at a convenient floor point.

For cleaning design, distance is only one variable. Frequency controls wavelength, and wavelength controls whether acoustic energy diffracts around obstructions or is shadowed. A lower-frequency horn may clean a remote surface better than a higher-output device at a less suitable frequency. The inverse-square law helps with open-air noise estimates, but plant cleaning decisions need geometry, deposit state and measured response.

Use with octave-band data

Distance calculations should be made by band where low-frequency acoustic cleaners are involved. A high-frequency component may attenuate strongly with distance and absorption, while a 63 Hz or 125 Hz component can remain prominent across a plant area. Barriers, lagging and cladding also perform differently by frequency. Applying one overall reduction to a mixed spectrum can therefore understate low-frequency impact.

For cleaning, the same issue appears inside equipment. The acoustic field is not just weaker with distance; its frequency content changes as surfaces absorb, reflect and scatter different wavelengths. A deposit behind a tube row may receive enough low-frequency energy even when a hand-held meter near the access door suggests a lower overall level. The inverse-square law gives a starting point, but commissioning should confirm the result with process trends and inspection rather than distance arithmetic alone.

Related terms

• Sound pressure level
• Attenuation (acoustic)
• Near field / far field