Question by Oh Shit It’s Thomas: What is the physical interpretation of a Laplace transformation?
I’ve come to understand that a Fourier transformation will take a periodic function from the time domain to the frequency domain. So what exactly does a Laplace transformation do?
Best answer:
Answer by Urmila
The Fourier transform analyzes the signal in terms of sinosoids, but the Laplace transform analyzes the signal in terms of sinousoids and exponentials. Traveling along a vertical line in the s-plane reveal frequency content of the signal weighted by exponential function with exponent defined by the constant real axe value. Traveling along a horisontal line reveal the exponential content of the signal weighted by sinousoids function with frequency defined by the constant imaginary axe value.In particular, traveling along the imaginaly axe reveal frequency content of the signal weighted by 1 which is equivalent to the Fourier transform of the signal.
Know better? Leave your own answer in the comments!
The other poster is exactly right between the fundamental differences being a translation of physical space/time into something in terms of sinusoids or exponentials.
I would not be concerned about this really, this is just a ‘transform.’ What I mean is, a more tractable example might be a transform in algebra, where we can use the logarithm “transform” to translate multiplication and division into addition and subtraction respectively in a “logarithm space.” We could then “invert” from logarithm space back to what we started with using its inverse (exponential). What the physical meaning of a logarithm does not make *too* much sense to ask. Same with integral transforms (which is what laplace and fourier transforms are). If you limit your interpretation to be something like a fourier transform taking something from the time domain to the frequency domain (and “signals”), that is restricting (although true for the most common of applications seen by undergraduates). Fourier transform has no restrictions on anything such as “time” being present in the integrals you are transforming. You can Fourier transform anything. Both the Laplace transform and Fourier transform provide “special” inner products (integral of a function over a defined interval weighted by (multiplied by) another, where you see the exponentials). These special inner products, maybe surprisingly, exhibit convenient properties in equation solving. Just like, possibly surprisingly, the logarithm is able to reduce multiplication/division to addition/subtraction.
In my discipline (plasma physics), we often have a differential equations that we aim to solve by doing a combination of transforms, which we choose to be the most convenient way to furnish a problem solution. That is to say, it is common language to say you Fourier-Laplace transform something. But evenstill, that definition is vague because you do not know what variables you are transforming with each. Contrary to the first part of what you wrote, I most often Fourier transform equations in *space*, not time, and further we usually Laplace transform in *time* not space. Why? It is all about boundary/temporal conditions. Laplace transforms of derivatives intrinsically involve initial conditions, Fourier transforms of derivatives do not. In plasma physics, given constraints of experiments, we often have initial (time) conditions, but not as often do we have boundary (space) conditions. So the ordering is largely based on this. We usually Fourier transform in space, Laplace transform in time, and solve the differential equation. Oftentimes, in literature for my field, results are quoted as fourier-laplace transforms, but you can also invert both transforms if the result is forgiving. Results quoted in terms of transforms are interpreted in their respective spaces, so you hone intuition to interpret these things, rather than need to invert back to get an equation in say x (space) and t (time). You mentioned Fourier transforms in time going to the frequency domain, you are certainly right in those applications. For us, we Fourier transform a spatial vector x into a “wave” vector k, and laplace transform in time t to a laplace transform variable (p, or s). Owing to the reasons mentioned by the other poster, you can see why people would commonly call a Fourier transform in space a “wave” vector since we translate according to a product with sinusoids. Ultimately though, these words are not the be all end all, this is just language. You choose an appropriate transform for the job you need to do.
So, there are many ways to interpret things, but the mathematics is what you are ultimately after when employing these methods. I would rather suggest you do not get hung up on frozen in interpretations like going to ‘frequency’ domain and view it as a tool, much like the logarithm, to simplify operations in a new space defined by the transform. A transform which you can invert if you like in the end after a solution is retrieved.