To make life easier.

(hard problem in time domain)---> transform to easier problem ----> solve easier problem -----> transform easier problem solution to solution of hard problem

When analysing data from dynamic events (time dependence histories) one often find broad banded (or narrow banded) energy contents distributed over a rich band of frequencies. If one needs to understand the underlying physics, one studies the importance of the energy contents at each frequency. Examples are fluid/free-surface processes, earthquakes, coupled systems with dampers.

You can also find some concrete examples, in a couple of papers I made available via research gate. These are;

1. sloshing motions in excited tanks;

2. numerical predictions of tuned liquid tank structural systems

In the latter paper the underlying idea is to imagine one have a structure with vibration problems - say it is easy to excite some on the first natural frequencies. On the structure is placed a water tank. The water in the tank will slosh from side to side when the structure vibrates and if the energy/frequency contents of excited water is tuned correctly it will suppress the vibration. In order to get this right one first vary the water depth and at each case study the couple system performance. For example, is the first natural sloshing frquencies near the natural frequencies of the structure, etc.

To understand Fourier transforms, I recommend to begin with the little easy to read book on "Sampling Theorem" by J. F. James. For real practice, matlab and simulink, are recommended. Note: it takes some practice to do right (the help function in matlab is also excellent).

Hope this is of some help.

Basically in the era when computers are not available, people have no choice but to transform a bunch of differential equations into another domain that can be easily manipulated and solved. However, in this era, your full spectrum of differential equations can be solved numerically, so you have a choice of not using these transforms. But I should say that for proofs, analytical manipulations, these transforms are still pretty important because this is the only way you can manipulate the equations to see trends and patterns and to get a feel of the dynamics.

@Yong, solution of differential equations by computers are based on certain algorithm, those are not analytical solution and even with the advent of most sophisticated computer, solution through numerical methods and/or by computer simulation are approximations one way or the other which may earn you a PhD but that's not all. In fact many have these wrong notion, that with computers you can replace mathematics, and I must say they miserably mistaken. Look at the many natural calamities, which need early warnings but we are still striving for any such full proof warning system. The reason is you need to have a analytical or theoretical benchmark for the solutions with numerical approaches which unfortunately for many cases we don't have. Analytical mathematics for the solutions differential equations has always been a challenge and will remain to be so. Although there are plenty of constraints (non-linearity and many more) and one should have patience, above all sound mathematical skill.

@Arghya, I am by no means belittling the importance of analytical solutions, but most of the real world problems are to date unsolvable analytically. While the mathematicians work on giving analytical and elegant solutions to difficult problems, we could use numerical methods to solve for the approximate solution of the problem so that at least we can start designing useful stuffs. In computational fluid dynamics, for example, analytical solutions are rarely found except for very few simplistic cases, and if we wait till some genius comes up with an analytical solution, most of the engineering problems would not be solved. Producing an analytical solution to difficult problems may earn you a Nobel Prize, but solving it numerically as it is now, gets the world going in many areas.

@Yong. You started with a humble note, but ended sarcastically by referring about Nobel Prize .... I belong to the analytical domain..... I did my PhD by solving only one problem..... I really thank god that my job is not determined by "publish or perish" theory. I have got very few works under my belt, but for that I never regret. Its not the Noble prize we are engaged in research but rather for the joy of the things, nothing else. Accolades come along the way if you are honest in whatever field you belong.... any way wish you all the best. World cannot only go with CFD or likewise numerical work and lastly who is genius and who is not is to be determined by "TIME", we are to little to comment on that. Best of luck ..... let's end this debate and concentrate on our own work.

@Arghya, I am vehemently against publishing for the sake of publishing, but let's just say that in every domain there are challenges, and we simply resort to different tools to solve meaningful problems in our area. I understand that different institutions may have different demands from their scientists and researchers and the one I am in is pretty much in the line of "publish or perish" business. Despite that, I think I am the few who do things for the sake of enjoyment. You may belong to a different school of thought than I am, but I think these shouldn't be a problem as all schools ought to compliment each other. Don't take it the wrong way, I have high regards to people who produces analytical solutions and I really think they are geniuses! I can assure you that the alleged sarcasm on my part does not stand in whatever way. Nobel prize - if you can produce a general analytical solution to the Navier-Stokes equation, Nobel prize would definitely be a side product. :)

You are welcome. yaah schools compliment that's true. Well I know the analytical solution of NS eqn is a millennium problem. One of the new 20 hardest problems of the Millennium. I'm not so talented to even to think about that. To be honest, for the last 5 years I have been trying hard to find a solution of a tsunami generation problem on a non-uniformly sloping beach. I'm probably almost at it. The non-uniformness is the key advancement here, analytically. These solutions, I believe will travel a long way in terms of raising the bar of the tsunami warning system. Last year I had the opportunity to talk on this at Athens, Greece, the greats in our field was there at the audience, they all encouraged me a lot but later I found some flaws in my solutions, hopefully these time I stand corrected. Anyway I welcome you to go through my papers they are there in public domain....Very few though...still. Ohh, let me tell you my initial work was on viscous fluid, finite depth ocean with/without boundary. I did a some important generalization on Cauchy-Poisson problem, a long paper. That was perhaps the seed of my future work on tsunami waves. Last but not the least, I went to your country last year at UTM, wooh its so big, a institute!!

Hello Arghya, nice work you have done. My boss is a fluid guy, so I know a little bit here and there about that. For myself, I worked on control system design in the past, improving existing predictive control algorithm for tackling nonlinear time-varying problems. Now I have moved on to solving population balance models for fermentation systems. I hope that I can provide a general framework to designing or steering bioreactor performance based on the regulatory nature of the microbial community itself. It's not a ground-breaking type of work, but at least for my country, I think many of the practical problems in the process industry are not given sufficient attention. Well, they may or may not adopt what I've done, but nonetheless, I do it for the sake of enjoyment. Somebody has to start the ball rolling before the need arises, right? You could come to University of Malaya on your next visit!

@Yong I wish you the best for all your present/future research endeavors.