1. **Problem Statement:** We want to prove the approximate wave speed formula $$v \approx \sqrt{\frac{gL}{2\pi}}$$ where $v$ is wave speed, $g$ is gravitational acceleration, and $L$ is wavelength. 2. **Formula and Explanation:** The exact wave speed for deep water waves is given by $$v = \sqrt{\frac{g\lambda}{2\pi} \tanh\left(\frac{2\pi d}{\lambda}\right)}$$ where $d$ is water depth and $\lambda$ is wavelength. For deep water, $d$ is large, so $\tanh\left(\frac{2\pi d}{\lambda}\right) \approx 1$, simplifying to $$v \approx \sqrt{\frac{g\lambda}{2\pi}}$$ 3. **Maclaurin Series for $\tanh x$:** Recall the Maclaurin series expansion for $\tanh x$ is $$\tanh x = x - \frac{x^3}{3} + \frac{2x^5}{15} - \cdots$$ 4. **Wave Speed Dependence on Depth:** Substitute $x = \frac{2\pi d}{\lambda}$ into the wave speed formula: $$v = \sqrt{\frac{g\lambda}{2\pi} \tanh\left(x\right)}$$ Using the series, $$\tanh(x) \approx x - \frac{x^3}{3} + \cdots$$ For shallow water ($d \ll \lambda$), $x$ is small, so $$v \approx \sqrt{\frac{g\lambda}{2\pi} \left(x - \frac{x^3}{3}\right)} = \sqrt{\frac{g\lambda}{2\pi} \left(\frac{2\pi d}{\lambda} - \frac{(2\pi d)^3}{3\lambda^3}\right)}$$ Simplify the first term: $$\frac{g\lambda}{2\pi} \cdot \frac{2\pi d}{\lambda} = gd$$ Higher order terms are small, so $$v \approx \sqrt{gd}$$ This shows wave speed depends mainly on water depth $d$ in shallow water. 5. **Computation for $d=0.5$ m and $L=200$ m:** Using deep water approximation: $$v \approx \sqrt{\frac{9.81 \times 200}{2\pi}} = \sqrt{\frac{1962}{6.283}} = \sqrt{312.3} \approx 17.67 \text{ m/s}$$ Using shallow water approximation: $$v \approx \sqrt{9.81 \times 0.5} = \sqrt{4.905} \approx 2.21 \text{ m/s}$$ Since $d=0.5$ m is much less than $L=200$ m, shallow water formula is more accurate here. 6. **Comment:** The wave speed formula $v \approx \sqrt{\frac{gL}{2\pi}}$ holds well for deep water waves where depth is large compared to wavelength. For shallow water, wave speed depends mainly on depth $d$ as $v \approx \sqrt{gd}$. This explains the physical behavior of waves in different water depths.