Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by ξ, η, and P (where dS is situated) by x, y, z.
Then
ρ² = (x − ξ)² + (y − η)² + z², ƒ² = x² + y² + z²;
so that
ρ² = ƒ² − 2xξ − 2yη + ξ² + η².
In the applications with which we are concerned, ξ, η are very small quantities; and we may take
| ρ = ƒ{ 1 − | xξ + yη | }. |
| ƒ² |
At the same time dS may be identified with dxdy, and in the denominator ρ may be treated as constant and equal to ƒ. Thus the expression for the vibration at M becomes
| − | 1 | ∫∫sin k { at − ƒ + | xξ + yη | } dxdy (1); |
| λƒ | ƒ |
and for the intensity, represented by the square of the amplitude,