We summarise Lamb’s treatment of a pressure disturbance on a free surface in a uniform stream, emphasising the contour‐integral evaluation of the far-field wavetrain, the underlying simple-harmonic free-surface condition, a Bessel–delta normalisation, and a geometric construction for envelopes of straight lines.
Pressure on a free surface in a uniform stream
We follow Lamb’s treatment of the pressure exerted on a free surface by a localised disturbance in a uniform stream, with a small linear damping included.
Notation and replacements
Relative to Lamb’s original notation we set
- \(\varpi \mapsto r\) (radius),
- \(\sigma \mapsto \omega\) (angular frequency),
- \(c \mapsto c_p\) (phase velocity),
- \(U \mapsto U\) (uniform background flow speed, used contextually),
and we keep \(g\) for gravity and \(\rho\) for density.
Assumptions (Lamb §242, p.398)
We assume:
- Steady motion: all fields are time-independent in the frame of the disturbance.
- Small dissipation: viscous/dissipative effects are modelled by a small linear drag.
- Irrotational flow in the bulk, so that a velocity potential exists.
Body-force model and basic framework
Let \(X,Y,Z\) be the components of the extraneous force per unit mass at \((x,y,z,t)\). Lamb models the effect of gravity and small linear damping by \[ X = -\mu(u - c_p), \qquad Y = -g - \mu v, \qquad Z = -\mu w, \] where:
- \((u,v,w)\) is the fluid velocity,
- \(c_p\) is a characteristic streamwise speed (used later in the dispersion relation),
- \(\mu\) is a small positive damping coefficient.
A canonical form of the free-surface elevation (for a particular forcing profile) can be written as \[ g \rho\, y = \kappa C \cdot \frac{(k - \kappa)\cos(kx) - \mu_1 \sin(kx)} {(k - \kappa)^2 + \mu_1^2}, \] where \[ \kappa = \frac{g}{c_p^2}, \qquad \mu_1 = \frac{\mu}{c_p}, \] and \(C\) encodes the strength of the applied pressure. This is Lamb’s p.439, eq. (22), recast in our notation.
Complexification and contour integral (Lamb §243)
Complex phase and wavenumber
We introduce complex parameters \[ c := \kappa + i\mu_1, \qquad \zeta := k + i m, \] with \(\kappa, \mu_1\) real. The denominator \[ (k - \kappa)^2 + \mu_1^2 \] can be recognised as \[ (k - \kappa)^2 + \mu_1^2 = (k - \kappa - i\mu_1)(k - \kappa + i\mu_1) = (k - c)(k - \bar{c}) = |k - c|^2. \]
For the numerator, we note \[ (k - \kappa)\cos(kx) - \mu_1 \sin(kx) = \Re\bigl[(k - \kappa - i\mu_1)e^{ikx}\bigr] = \Re\bigl[(k - c)e^{ikx}\bigr]. \]
Thus the integrand can be reformulated in terms of \[ \frac{e^{ikx}}{k - c}, \] and after extending \(k\) to the complex plane the basic integral becomes of the form \[ \int \frac{e^{ix\xi}}{\xi - c}\,d\xi. \]
This is the analytic kernel to which we apply contour integration.
Why a factor \(-2\pi i\)?
We consider a contour that consists of a large semicircle together with a small circle around the pole at \(\xi=c\). Suppose we take the total contour \(C_{\mathrm{total}}\) in the positive (counterclockwise) sense, enclosing the pole at \(\xi=c\). By the residue theorem, \[ \oint_{C_{\mathrm{total}}} f(\xi)\,d\xi = 2\pi i\,\mathrm{Res}(f,\xi=c). \]
However, in the picture used by Lamb the small circle around the pole is traced clockwise, opposite to the positive orientation. If we call this inner circle \(C_{\mathrm{inner}}\), then \[ \int_{C_{\mathrm{inner}}(\text{clockwise})} f(\xi)\,d\xi = -\int_{C_{\mathrm{inner}}(\text{counterclockwise})} f(\xi)\,d\xi = -\bigl[2\pi i\,\mathrm{Res}(f,\xi=c)\bigr]. \]
Hence the contribution from the small circle appears with a factor \(-2\pi i\) times the residue. This sign is purely an orientation effect.
Jordan’s lemma and the real-axis integral
We now write the contour integral as \[ \oint \frac{e^{ikx}}{k - c}\,dk = \int_{-\infty}^{\infty}\frac{e^{ikx}}{k - c}\,dk + \int_{C_1}\frac{e^{ikx}}{k - c}\,dk, \] where \(C_1\) is the large semicircle in the upper half-plane (for \(x>0\)) or lower half-plane (for \(x<0\)).
For \(x>0\), Jordan’s lemma implies that the integral over the upper semicircle \(C_1\) vanishes: \[ \int_{C_1}\frac{e^{ikx}}{k - c}\,dk \to 0 \quad (x>0), \] provided the integrand decays sufficiently fast on the arc. If the chosen contour does not enclose the pole at \(k=c\), then by Cauchy’s theorem \[ \oint \frac{e^{ikx}}{k - c}\,dk = 0, \] and the real-axis integral is then determined by whatever small-arc contributions we include (e.g. when detouring around the pole or taking limits).
When \(m<0\) and \(\kappa,\mu_1>0\), Lamb obtains the identity \[ \int_0^\infty \frac{e^{ikx}}{k - (\kappa + i\mu_1)}\,dk = \int_0^\infty \frac{e^{-ikx}}{k + (\kappa + i\mu_1)}\,dk, \] reflecting the analytic structure and the absence of enclosed poles for the chosen contour.
Lamb’s formulae and physical interpretation
For \(x>0\), Lamb shows that (his eq. (25)) can be written as \[ \begin{aligned} \frac{\pi g\rho}{\kappa P}\,y(x) &= -2\pi e^{-\mu_1 x}\sin(\kappa x) + \int_0^\infty \frac{(k+\kappa)\cos(kx) - \mu_1\sin(kx)} {(k+\kappa)^2 + \mu_1^2}\,dk \\ &= -2\pi e^{-\mu_1 x}\sin(\kappa x) + \int_0^\infty \frac{(m - \mu_1)e^{-mx}}{(m - \mu_1)^2 + \kappa^2}\,dm. \end{aligned} \tag{25} \]
For \(x<0\), the corresponding expression (his eq. (26)) is \[ \frac{\pi g\rho}{\kappa P}\,y(x) = \int_0^\infty \frac{(m + \mu_1)e^{mx}}{(m + \mu_1)^2 + \kappa^2}\,dm. \tag{26} \]
Interpretation.
- The first term in (25) represents a downstream train of simple-harmonic waves of wavelength \[ \lambda = \frac{2\pi c_p^2}{g}, \] with amplitude decaying as \(e^{-\mu_1 x}\).
- The integral terms in (25)–(26) represent the nearfield deformation of the free surface, which is large only near the disturbance and decays rapidly as \(|x|\to\infty\), even for very small \(\mu_1\).
In the limit \(\mu_1\to 0\), Lamb’s expressions simplify to \[ \frac{\pi g\rho}{\kappa P}\,y(x) = -2\pi\sin(\kappa x) + \int_0^\infty \frac{\cos(kx)}{k+\kappa}\,dk = -2\pi\sin(\kappa x) + \int_0^\infty \frac{m e^{-mx}}{m^2 + \kappa^2}\,dm, \] exhibiting a pure sinusoidal far-field plus a rapidly decaying remainder.
Simple-harmonic free-surface condition (Lamb §430)
For simple-harmonic motion with time factor \(e^{i(\sigma t + \epsilon)}\), the free-surface condition in deep water takes the form \[ \sigma^2 \phi = g\,\frac{\partial \phi}{\partial z} \quad\text{on } z=0, \tag{4.1} \] where \(\phi\) is the velocity potential and \(\epsilon\) is the initial phase of the wave. This relation encapsulates the balance between vertical velocity and free-surface displacement for a linear gravity wave.
Bessel normalisation and the Dirac delta (Lamb §430, eqs. (12)–(13))
We consider the normalisation condition \[ \int_0^\infty f(\alpha)\,J_0(k\alpha)\,\alpha\,d\alpha = 1. \tag{5.1} \]
If we choose \[ f(\alpha) = \frac{\delta(\alpha)}{2\pi \alpha}, \tag{5.2} \] i.e. the two-dimensional Dirac delta in polar coordinates, then \[ \int_0^\infty f(\alpha)\,2\pi\alpha\,d\alpha = \int_0^\infty \frac{\delta(\alpha)}{2\pi\alpha}\,2\pi\alpha\,d\alpha = \int_0^\infty \delta(\alpha)\,d\alpha = 1, \] so the normalisation is consistent.
Substituting \(f(\alpha)\) into (5.1) gives \[ \int_0^\infty f(\alpha)\,J_0(k\alpha)\,\alpha\,d\alpha = \int_0^\infty \frac{\delta(\alpha)}{2\pi\alpha} J_0(k\alpha)\,\alpha\,d\alpha = \frac{J_0(0)}{2\pi} = \frac{1}{2\pi}, \] using \(J_0(0)=1\). This illustrates how the polar delta normalisation introduces a factor \(1/(2\pi)\) in the corresponding Bessel integral.
An example of pressure distribution (Lamb §467, eq. (20))
Lamb gives a typical example of a one-dimensional pressure distribution on the free surface as \[ p'(x) = \frac{P}{\pi}\,\frac{b}{b^2 + x^2}, \tag{6.1} \] a Lorentzian profile of half-width \(b\). This form is often used as a smooth model for a localised pressure patch or load.
Envelopes of a family of straight lines (Lamb §469)
Family of lines
We consider the family of straight lines \[ x\cos\theta + y\sin\theta = p(\theta), \tag{7.1} \] where:
- \(\theta\) is the angle that the line’s normal makes with the \(x\)-axis,
- \(p(\theta)\) is the perpendicular distance from the origin to the line.
As \(\theta\) varies, (7.1) describes a one-parameter family of lines. The envelope of this family is a curve \(C\) that is tangent to each member of the family for exactly one value of \(\theta\).
Envelope condition via differentiation
If \((x(\theta),y(\theta))\) is the point of tangency on the line corresponding to \(\theta\), then it must satisfy both:
- The line equation itself: \[ x(\theta)\cos\theta + y(\theta)\sin\theta = p(\theta), \tag{7.2} \]
- The tangency (envelope) condition, obtained by differentiating (7.1) with respect to \(\theta\) and requiring that the change still vanish at the contact point: \[ \frac{d}{d\theta}\bigl[x(\theta)\cos\theta + y(\theta)\sin\theta - p(\theta)\bigr] = 0. \tag{7.3} \]
Expanding (7.3), we obtain \[ x'(\theta)\cos\theta - x(\theta)\sin\theta + y'(\theta)\sin\theta + y(\theta)\cos\theta = p'(\theta), \tag{7.4} \] where primes denote derivatives with respect to \(\theta\).
Solving the system (7.2)–(7.4) for \(x(\theta),y(\theta)\) yields the envelope. Eliminating \(x'(\theta)\) and \(y'(\theta)\), we find the familiar parametric representation: \[ \boxed{ \begin{aligned} x(\theta) &= p(\theta)\cos\theta - p'(\theta)\sin\theta,\\ y(\theta) &= p(\theta)\sin\theta + p'(\theta)\cos\theta. \end{aligned} } \tag{7.5} \]
These formulae express the envelope curve in terms of the distance function \(p(\theta)\) and its derivative, and they are widely used in geometric optics and wavefront constructions, including the analysis of caustics in ship-wave patterns.