Speed, wavelength, and range — the acoustic ruler

The speed of sound is not constant

The speed of sound in seawater is not a fixed 1500 m/s. In practice it depends on temperature, salinity, and pressure (depth). Temperature dominates near the surface, while pressure becomes more important with increasing depth. In most of the open ocean, salinity tends to contribute less than temperature or pressure.

For an introduction, using a representative value near 1500 m/s is enough for the calculations. For precise ranging or when you need to account for refraction, however, you need a full sound-speed profile.

The Mackenzie sound-speed approximation

The simulator in this course and the minimum implementation in Chapter 7 both compute the speed of sound c from temperature, salinity, and depth using the Mackenzie (1981) 9-term approximation. The representative 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

Here T is temperature [°C], S is salinity [PSU], and D is depth [m]. The higher-order T terms capture the non-linear temperature dependence near the surface, and the D terms capture pressure dependence at depth. The salinity coefficient is just 1.34, which is small compared to the temperature coefficient 4.59 — the equation itself shows that salinity contributes relatively little. The Chapter 6 simulator uses this same equation to compute the speed of sound when you move the temperature, salinity, and depth sliders.

Read wavelength as c / f

Wavelength is the length of one cycle: λ = c / f. The higher the frequency, the shorter the wavelength. Short wavelengths make it easier to resolve fine structure, but as we will see later, absorption also grows.

For example, with c = 1500 m/s and f = 30 kHz, the wavelength is 0.05 m — that is, 5 cm. Wavelength is also the yardstick for choosing array element spacing.

Convert round-trip time back to range

An active sonar measures the round-trip time from transmission to reception of the echo. One-way range is c × t / 2. What matters is keeping the units of the speed c and the time t consistent. Using m/s and s gives you meters directly.

The equation is simple, but it carries a lot of practical intuition. Double the range, and the time also doubles. Changing the frequency does not change the basic timing equation as long as c stays the same. Once those correspondences are internalized, the numbers in the simulator become much easier to read.

Key takeaways from this chapter

• The speed of sound depends on temperature, salinity, and pressure. In the open ocean, salinity contributes relatively little.
• Wavelength is λ = c / f. Higher frequency means shorter wavelength.
• The equation for range from round-trip time is distance = c × t / 2.