Sound Propagation Beneath the Waves

Acoustic waves are the primary means of long‑distance communication and sensing in the ocean. Light attenuates within a few tens of meters and radio waves fare even worse, leaving sound as the only practical tool for submarines, underwater vehicles, and marine mammals to probe their surroundings. As a sound wave travels through seawater, its intensity diminishes due to two principal mechanisms: geometric spreading of the wavefront and absorption of acoustic energy by the water itself. The cumulative effect of these processes is called transmission loss, commonly expressed in decibels (dB). Understanding transmission loss is crucial for tasks such as estimating sonar performance, planning underwater acoustic communication links, or assessing the range at which marine life might be impacted by anthropogenic noise.

The calculator above combines a simple geometric spreading model with the well‑known Thorp absorption formula to estimate transmission loss for a continuous tone. You specify the range from the source to the receiver in kilometers, the signal frequency in kilohertz, and whether the sound spreads spherically or cylindrically. Spherical spreading assumes sound radiates equally in all directions, causing intensity to drop with the square of distance. Cylindrical spreading represents propagation constrained by boundaries such as the sea surface and seafloor, leading to a slower decline with range. In reality, oceanic environments transition between these extremes depending on depth, channeling effects, and seafloor properties, but considering both models offers useful bounds.

The total transmission loss is computed as the sum of geometric spreading and absorption. For spreading we use , where is range in meters and equals 20 for spherical spreading or 10 for cylindrical spreading. Absorption is handled by the Thorp equation, appropriate for frequencies between a few hundred hertz and hundreds of kilohertz:

,

where frequency is in kilohertz and the resulting absorption coefficient is in dB/km. This empirical formula encapsulates multiple physical processes including viscosity, ionic relaxation, and boric acid and magnesium sulfate absorption. Multiplying by range gives the absorption loss component. The calculator sums this with the spreading loss to give the total transmission loss in decibels.

Why are these calculations useful? Consider designing an acoustic modem to transmit data between an underwater glider and a surface buoy. Knowing the transmission loss allows engineers to determine the required source level to achieve a target signal‑to‑noise ratio at the receiver. In naval operations, predicting transmission loss helps estimate the detection range of a sonar system or assess how readily a submarine might be heard by adversaries. Marine biologists use similar assessments to evaluate how far industrial noise will propagate and potentially disturb sensitive species.

The Thorp formula captures a characteristic frequency dependence: absorption is minimal at low frequencies but rises rapidly above about 10 kHz. At 1 kHz, α is only around 0.07 dB/km, so a wave can travel many tens of kilometers before absorption becomes significant. At 100 kHz, α exceeds 30 dB/km, confining high‑frequency signals to short ranges. The table below highlights this trend by listing absorption coefficients computed from the Thorp formula at several representative frequencies.

Frequency (kHz)	Absorption α (dB/km)
0.5	0.05
1	0.07
5	0.46
10	1.1
50	11
100	34

The frequency dependence has practical consequences for communication system design: low frequencies travel farther but carry less bandwidth, while high frequencies permit faster data rates but suffer greater loss. Engineers often employ spread‑spectrum modulation, error‑correcting codes, or adaptive power control to cope with these limitations.

Transmission loss also varies with environmental conditions. Temperature, salinity, and pressure influence the speed of sound, which in turn refracts acoustic rays and can create sound channels where energy is trapped and guided over enormous distances, as exemplified by the deep sound channel used by whales and long‑range sonar. Scattering from bubbles, turbulence, and seafloor roughness adds further complexity. The simple calculator here does not account for these factors, but it provides a foundational estimate that can guide more detailed modeling.

To use the calculator, enter the range between your source and receiver, the acoustic frequency of interest, and select a spreading model. The script converts the range to meters for the geometric term, computes the absorption coefficient with the Thorp formula, and then sums the two contributions. The output is the transmission loss in decibels. For example, a 10 kHz signal traveling 20 km with spherical spreading experiences approximately 20 log₁₀(20,000) ≈ 86 dB of spreading loss plus 1.1 × 20 = 22 dB of absorption, totaling about 108 dB. Such a path would require a source level more than one hundred billion times stronger than the received level.

Even at shorter ranges the ocean can be surprisingly harsh on sound. At 100 kHz, the absorption loss alone is roughly 34 dB per kilometer, meaning the intensity drops by a factor of 2,500 after just one kilometer. This explains why high‑resolution imaging sonars used for near‑field inspections operate at such high frequencies yet are effective only at short ranges.

The ability to reason about transmission loss empowers scientists and engineers to design systems that respect both technical requirements and environmental constraints. Whether you are exploring the feasibility of an underwater acoustic link, estimating detection ranges for sonar, or assessing the potential impact of a proposed marine construction project, a clear understanding of sound attenuation is essential. Experiment with different ranges, frequencies, and spreading assumptions in this calculator to build intuition about how underwater sound behaves.