how to obtain \(p=a \cos ^2 \theta\)
Question:
\(p=a \cos ^2 \theta,\)
In this context, \(a\) is simply the constant of integration (or “crest‑label”) that picks out one particular wave‐ridge in the pattern. Equivalently you can say:
- For each fixed wave‐crest (the “\(m\)th” ridge behind the moving pressure‐point), the ratio \[ \frac{p}{\lambda} \]
is the same for all directions \(\theta\), and differs by exactly 1 as you go from one ridge to the next. Call that constant ratio \(m\).
- Hence for the \(m\)th crest \[ p \;=\; m\,\lambda \quad\Longrightarrow\quad a \;=\; m\,\lambda\bigl|_{\theta=0}, \]
so that \(a\) is precisely the value of \(p\) when \(\theta=0\). In other words, it is the distance from the ship (or pressure‐point) to that particular crest straight astern .
Derivation of \(p = a\cos^2\theta\)
- Dispersion relation for infinite depth
In deep water one shows (equation (24) in your excerpt) that \[ c^2\cos^2\theta \;=\; V^2 \;=\;\frac{g\,\lambda}{2\pi} \quad\Longrightarrow\quad \lambda \;=\; \frac{2\pi\,c^2}{g}\,\cos^2\!\theta. \]
- Constant ratio \(p/\lambda\)
Stationary‐phase in the position variable \(Q\) tells us that \[ \frac{p}{\lambda} \;=\; m \quad\text{(an integer labeling the crest).} \]
Hence
\[ p \;=\; m\,\lambda \;=\; m\;\frac{2\pi\,c^2}{g}\;\cos^2\!\theta. \]
- Defining \(a\)
We set \[ a\;=\;m\,\frac{2\pi\,c^2}{g} \;\;=\;\; p\bigl|_{\theta=0}, \]
so that finally
\[ p = a\cos^2\!\theta\quad\bigl(\text{for infinite depth}\bigr). \]
Physical meaning of \(a\)
Crest‑number : As you move from one crest to the next, \(m\to m+1\), so \(a\) increases in steps of \(\tfrac{2\pi c^2}{g}\).
Geometry : \(a\) is exactly the radial distance from the ship to that crest along the line \(\theta=0\) (straight behind).
Scale : In deep water the basic length scale \(\tfrac{2\pi c^2}{g}\) often appears in ship‐wave theory; \(a\) is just an integer multiple of that, selecting which ridge you’re drawing.
So nothing more exotic was intended by the symbol \(a\)—it’s just the constant you get when you enforce “\(p/\lambda=\) integer,” and geometrically it is the intercept of the chosen crest on the ship’s track (\(\theta=0\)).
why does \(p=m\lambda\) indicates stationary phase?
When we carry out the stationary-phase approximation in the spatial integral over all source points \(Q\), we look for those \(Q\) for which the phase of the integrand
\[ \phi(Q)=k(V t(Q)-\varpi(Q)) \]
is both
Stationary in \(Q\left(\nabla_Q \phi=0\right)\) - this picks out the ray direction and gives \(V=c \cos \theta\).
Constructively interfering, which means that at those stationary points the phase itself must be an integer multiple of \(2 \pi\).
(i) Stationary condition in \(Q\) \[ \frac{\partial \phi}{\partial Q}=0 \quad \Longrightarrow \quad \dot{\varpi}=V \quad \Longrightarrow \quad V=c \cos \theta \]
This fixes the direction \(\theta\) along which contributions from nearby \(Q\) 's add coherently.
(ii) Constant-phase (crest) condition
Once we've found the stationary point, the peaks in the physical wave-pattern occur where the phase itself is a constant multiple of \(2 \pi\). Denote that integer by \(m\). Then
\[ \phi=k(V t-\varpi)=k p=2 \pi m \quad \Longrightarrow \quad k p=2 \pi m \quad \Longrightarrow \quad \frac{p}{\lambda}=m, \]
since \(k=2 \pi / \lambda\). That is exactly the statement
\[ \frac{p}{\lambda}=m \quad(\text { an integer labeling the } m \text { th crest }) . \]
In plain language - Stationary-phase tells you where in space (which \(Q\) ) the waves line up to give a strong signal at \(P\). - The integer-multiple-of- \(2 \pi\) part tells you which crest we're looking at: successive crests differ by a phase jump of \(2 \pi\), so their path-difference \(p\) differs by one wavelength. - Putting those together yields
\[ p=m \lambda \quad \Longrightarrow \quad \frac{p}{\lambda}=m \]