Sound Speed, Wavelength, and Travel Time — The Basics of Range

Sound speed in seawater varies by location

The speed of sound in seawater is not always 1500 m/s. It changes mainly with temperature, salinity, and pressure (depth). In general, temperature has a strong effect near the surface, while the effect of pressure becomes more pronounced with increasing depth. In most open-ocean settings, salinity has a smaller effect than temperature or pressure.

For the calculations in this chapter, we use the representative value c = 1500 m/s. This approximation is sufficient for learning the basics. To calculate range more precisely or account for refraction, where sound bends as it travels, you need a sound-speed profile that shows how sound speed changes with depth.

The Mackenzie sound-speed approximation

The Chapter 6 simulator and the Chapter 7 implementation example use the Mackenzie (1981) 9-term approximation to calculate sound speed c from temperature, salinity, and depth. A common form of the equation is:

c [m/s] = 1448.96
        + 4.591 T − 5.304×10⁻² T²    + 2.374×10⁻⁴ T³   ← temperature terms
        + 1.340 (S − 35)                                ← salinity term
        + 1.630×10⁻² D + 1.675×10⁻⁷ D²                  ← depth terms
        − 1.025×10⁻² T (S − 35) − 7.139×10⁻¹³ T D³      ← interaction terms

T is water temperature [°C], S is salinity [PSU], and D is depth [m]. The quadratic and cubic T terms show that the relationship between temperature and sound speed is not a simple straight line, while the D terms represent the effect of increasing pressure with depth. The salinity-term coefficient is 1.34, and the linear temperature-term coefficient is 4.59, but the actual contribution of each factor depends on the values of T, S, and D, not on the coefficients alone. Chapter 6 recalculates sound speed with this equation whenever you move a slider.

Find wavelength by dividing sound speed by frequency

Wavelength is the distance a wave travels during one cycle. If c is sound speed and f is frequency, then λ = c / f. At the same sound speed, a higher frequency gives a shorter wavelength. A short wavelength can help reveal finer differences, but higher-frequency sound is also absorbed more readily by seawater.

For example, if c = 1500 m/s and f = 30 kHz, the wavelength is 0.05 m, or 5 cm. Wavelength is also an important reference when deciding how far apart to place multiple receiver elements.

Convert round-trip time to one-way range

Active sonar measures the round-trip time t from transmitting sound to receiving its echo. The one-way range to the target is c × t / 2. If sound speed c is in m/s and time t is in s, the result is in meters.

We divide by 2 because the measured time includes both the outward and return trips. At the same sound speed, doubling the range doubles the round-trip time. Changing frequency does not change this basic relationship between range and time. Once you understand that relationship, the values in the Chapter 6 simulator become easier to interpret.

Chapter summary

• Sound speed in seawater changes with temperature, salinity, and pressure.
• Wavelength is λ = c / f; at the same sound speed, a higher frequency gives a shorter wavelength.
• distance = c × t / 2 converts round-trip time into one-way range.