A water wave of length L meters in water of depth d meters...

A water wave of length $L$ meters in water of depth $d$ meters has velocity $v$ satisfying the equation $$v^{2}=\frac{4.9 L}{\pi} \frac{e^{2 \pi d / L}-e^{-2 \pi d / L}}{e^{2 \pi d / L}+e^{-2 \pi d / L}}$$ Treating $L$ as a constant and thinking of $v^{2}$ as a function $f(d)$ use a linear approximation to show that $f(d) \approx 9.8 d$ for small values of $d$. That is, for small depths, the velocity of the wave is approximately $\sqrt{9.8 d}$ and is independent of the wavelength $L$.

A water wave of length $L$ meters in water of depth $d$ meters has velocity $v$ satisfying the equation
$$v^{2}=\frac{4.9 L}{\pi} \frac{e^{2 \pi d / L}-e^{-2 \pi d / L}}{e^{2 \pi d / L}+e^{-2 \pi d / L}}$$
Treating $L$ as a constant and thinking of $v^{2}$ as a function $f(d)$ use a linear approximation to show that $f(d) \approx 9.8 d$ for small values of $d$. That is, for small depths, the velocity of the wave is approximately $\sqrt{9.8 d}$ and is independent of the wavelength $L$.

Key Concepts

Taylor Series Expansion

The Taylor series expansion is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For small deviations from this point, the series can be truncated to its first term, providing a linear (or higher degree) approximation. This plays a critical role when approximating functions for small arguments, as in the case of expanding functions like the hyperbolic tangent.

Linear Approximation

Linear approximation is a method in calculus where a function is approximated by its tangent line at a given point. This technique relies on the idea that for values very near the point of tangency, the function behaves almost linearly, making the first derivative a good measure of change. It is especially useful when dealing with complex functions, as it simplifies analysis and computation in a local region.

Hyperbolic Functions

Hyperbolic functions, such as the hyperbolic sine and cosine, arise naturally when dealing with exponential expressions and have properties analogous to the trigonometric functions. In particular, the hyperbolic tangent function, which is the ratio of the hyperbolic sine to the hyperbolic cosine, has a simple linear behavior for small argument values. This property is instrumental in approximating wave speeds in shallow water scenarios.

Shallow Water Wave Approximation

The shallow water wave approximation is a concept in fluid dynamics that applies when the depth of the water is small compared to the wavelength. Under this condition, the wave speed becomes primarily dependent on the depth rather than the wavelength. This simplification often results from linearizing the governing equations, allowing complex relationships to be reduced to simpler forms like a square-root dependence on depth.

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