What is the best way to accurately measure wavelength between crests from a pier?


One simple way is to measure the period $T$ by timing the arrival of each crest and estimate the wave speed $c$ from the depth of the water column $d$. Then one can compute a wavelength $\lambda$.

Assuming the waves are shallow-water waves ($d/\lambda < 1/11$) and linear (modeled by a sine or cosine function), the error should be no more than 10% (is that accurate enough? It gets more complicated otherwise).

Since we assumed these are shallow water waves they are nondispersive and the speed is simply $c = \sqrt{gd}$, where $g$ is the gravitational acceleration.

Thus, the wavelength is $$\lambda = c\ T=\sqrt{gd}\ T\,.$$

  • $\begingroup$ What's the best tools I can bring with me to help measure everything? $\endgroup$
    – Joe Shmoe
    Jul 20 '15 at 5:50
  • $\begingroup$ You could measure the depth of the water by dropping a weighted line from the pier to the top of the water, measure the distance & then drop the weight to the bottom of the water & measure the distance. Subtracting the two distances gives the depth of the water. $\endgroup$
    – Fred
    Jul 20 '15 at 5:50
  • $\begingroup$ Yes, this is the simplest method I can think of that does not involve sophisticated instruments. $\endgroup$ Jul 20 '15 at 15:20
  • $\begingroup$ +1 As this is the logical approach to answering the question. However, I'll note that the method outlined above makes the assumption that the waves under consideration follow the linear dispersion relation. Its relevance to your measurement depends on a variety of factors, but establishing this relationship in nature is not necessarily trivial, as its applicability to your scenario needs to be established. The way one could do this is by using a video camera (or even better, 2 cameras) and connecting spatial and temporal information. $\endgroup$
    – Nick P
    Jul 21 '15 at 4:43

Ideally you'd like to avoid having to estimate anything.

Depending on the shape and orientation of the pier relative to the waves, you might be able to find some volunteers (at least 1 anyway) and space them out at approximately the right distance. Then get them to raise a hand when a crest passes them. Adjust their spacing until they are raising their hand at the same time. If the crests are simultaneous and there is only one trough between them, they are one wavelength apart.

You might need to adjust for the angle of the pier, for example in this case, wavelength $\lambda$ is related to the inter-volunteer distance $d$ and pier angle $\theta$ as $\lambda = d \cos\theta$

measuring wavelength from a pier

Measure as many crests as you can, keeping track of the tide. As Isopycnal Oscillation explained in another answer, the wavelength with vary with water depth. Perhaps measure around high tide, and again around low tide. Take the means of your various sets of measurements.

You'll need a longish tape measure, or phones with GPS, and a stopwatch, or phone with clock, if you want to measure periods as well (you might as well).

  • $\begingroup$ -1 Because the OP asks for an "accurate" method. :) In shallow water, measuring period is the direct way to obtain wavelength, is easier to measure, and does not require human resources. Averaging between low and high tide would not yield a measurement of any specific wave, especially if the incoming wave field changes over the period of several hours. $\endgroup$ Jul 20 '15 at 15:17
  • $\begingroup$ Measuring the period and inverting for wavelength is not direct. It requires an estimate of wave speed. So I guess I don't buy that it's easier to time arrivals and invert for wavelength. Besides, the time invariance is part of the interest I think... and of course you have that problem however you do it. Part of the problem is that it's hard to measure nature! $\endgroup$
    – kwinkunks
    Jul 20 '15 at 15:28

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