You must define “deviation” more precisely in order to measure it.

For instance, it might be the deviation in one or more paths (rays) that sound radiated by a single source follows in the sea (generally not straight owing to changing refraction of sound in the sea).

Or deviations in shadow or convergence zones.

Or it might be the deviation in the intensity of the sound field at selected points owing to changing concentration or spreading of ray boundaries, or changing interference patterns when the sound follows multiple paths from source to receiver.

There are other possible deviations as well. Each requires a different experimental method to observe and quantify in practice.

Agreed with Ronald completely - further definition of your problem is needed.

Yet another interpretation _could_ be - how does the shape of an acoustic pulse change as a result of dispersive propagation (where different frequencies travel at different speeds)

Thanks a lot guys.....my question is about deviation of sound rays from its normal propagation path(the first one Mr. Kessel mentioned). Let me add a little more. All i have is salinity, temperature and pressure data.I can calculate speed of sound at any particular part by using pressure, temperature and salinity recorded in that part. But how much sound ray is deviated as a result of change in speed of sound is my concern. In short, is there any mathematical relationship between change in the speed of sound and the resultant deviation?

As you know, ray paths

1. follow straight lines wherever the gradient of the sound speed is zero (constant sound speed);

2. bend wherever the gradient of the sound speed is non-zero (changing sound speed);

3. bend smoothly wherever the gradient of the sound speed is non-zero and finite (smoothly changing sound speed);

4. bend discontinuously wherever the gradient of the sound speed in infinite (at a reflective boundary or interface between distinct layers.

The key parameter in all cases is the gradient of the sound speed (emphasis on gradient). So the relationship that you want to explore quantitatively is the relationship between the 1) gradient of the sound speed, and 2) degree of ray bending.

Presumably you are most interested in case (3) above, smooth ray bending.  A simple example for mathematical analysis occurs when the gradient of the sound speed c is constant in water depth z (dc/dz = constant, not equal to zero), and zero in the horizontal direction (when there is no horizontal ray bending). Then it can be shown that the radius of curvature R of the ray path in a vertical plane is

  R = - 1 / (dc/dz X ray_constant)

Where dc/dz is the vertical gradient of the sound speed c, z is positive downwards (i.e., z increases with depth), and the ray_constant is constant at all points along the ray,

  ray_constant =  cosine(theta(z)) / c(z)

   theta(z) = angle of the direction of the ray path; with theta(z) = 0 degrees for a horizontal ray

  c(z) i= vertical sound speed profile as a function of depth z

These equations can be found in equations in Hovem’s article in link provided below (see his equations (1) to (4))

More generally, for other kinds of vertical sound speed profiles c(z) that have changing vertical gradient dc/cz, you may need a

1. ray-trace programme to see and quantify the effect of changing sound speed on ray bending (mathematics of general ray bending are also described by Hovem). Ray-trace programmes that you might consider using can be found for download in the second link provided below; and you may need to

2. develop a measure of the overall measure of the effect, degree, impact, or sensitivity of ray bending which, when applied to the outcome of ray-traces forecast by the ray-trace model, assesses the deviation of ray paths to particular changes in the gradients of sound-speed profiles, in a way that is especially suited to your particular application.

Ronald

http://cdn.intechopen.com/pdfs-wm/45578.pdf

http://oalib.hlsresearch.com/Rays/

Thanks alot Mr. Kessel the detailed answer....I am extremely grateful.

I would suggest to use a software like AcTUP: http://cmst.curtin.edu.au/products/actoolbox.cfm

I'd also suggest to visit http://oalib.hlsresearch.com

Gianni

I'm agree with Mr.Gianni Pavan