I think that question is not fully correct... Immagine the simple wave equation

df/dt + u df/dx = 0 (u=constant)

in a periodic domain with some initial function f0(x).

Now, you can see that, according to a Fourier series, a real solution f(x,t) can be writtes as a sum of products of two complex Fk(t)*exp(i*k*x). By substituting it in the equation, you can easily see that Fk(t)=Fk(0)*exp(-i*u*t).

The key is that using the complex nomenclature is nothing but a way to rewrite a real function (consider the conjugate part in the series).

Purvi,

It makes the mathematics a lot easier when speaking of oscillatory functions - one could write down the transverse components of a rotating function:

x = x0*sin(wt)

y = y0*cos(wt)

but handling the function, (and any displacements) become simply more compact if we map 'y' to the imaginary axis and the function then becomes a complex number.

Also, just in one plane, x = A*cos(wt) + B*sin(wt) is a general sine-shaped wave which could also be described as x = C*cos(wt+d). This is the real part of C*exp(j*(wt+d)). Using complex maths often simplifies the equations and can make it easier to visualise what is happening (the projection of a rotating vector onto the real line). The complex part probably has no physical meaning - but you never know - maths has an annoying habit of turning out to represent reality.

I think that question is not fully correct... Immagine the simple wave equation

df/dt + u df/dx = 0 (u=constant)

in a periodic domain with some initial function f0(x).

Now, you can see that, according to a Fourier series, a real solution f(x,t) can be writtes as a sum of products of two complex Fk(t)*exp(i*k*x). By substituting it in the equation, you can easily see that Fk(t)=Fk(0)*exp(-i*u*t).

The key is that using the complex nomenclature is nothing but a way to rewrite a real function (consider the conjugate part in the series).

The imaginary part has no physical meaning. The wave is a real entity. It is modeled by a real equation : A*cos(kx-wt), which is the real part of A*exp i(kx-wt). The complex representation is more suitable (easier) when manipulating and developing heavy mathematical equations than trigonometric equations. See, for example, the Poynting's vector calculation where the conjugate form is introduced :

exp i(kx-wt)* exp (-)i(kx-wt)=1, but cos(kx-wt)*cos(kx-wt) does not.

There is a specific case when the analysis of the imaginary part has a "physical" counterpart: the numerical solution of the wave equation.

Introducing a discretization produces the so-called "modified wavenumber". In case it has an imaginary part (as happens for upwind schemes) it addresses the presence of a numerical viscosity that smoothes the amplitude of the wave.

thank you very much

regards,

purvi

I add one thing to all above said explanations, It is that we are all aware of the fact that a WAVE can never be seen. Whether its a Sound/Light/Thermal/Electrical/Gravitational/Matter to mean to say any kind of wave is not visible, but their Effects can be perceived.Thus the Presence/Existence of WAVE is not in real plane/world(As it can't be seen ) They all lie in the Complex Plane/Space/World. Hence forth to represent their presence in complex plane, we use i, the imaginary number with the position & very intelligently to reveal its effect, we associate real quantity w i.e angular velocity with time.

Matter waves can be seen. Waves on a rope and on the sea are clearly visible, and weight and spring structures can be built to carry compression waves that are big enough and slow enough to see. The displacements and the speed of movement of individual parts in these waves are both real quantities that can be seen.

Yes, many waves can be seen. A chord shows waves, a earthquake show waves, too...

I think we see the effect of the wave on the rope or in water rather than the wave itself. The different parts of the rope do not move but just oscillate in their positions causing the energy to be transmitted from a part to the neighboring one. i.e. the rope is just responding to the wave excitation : the energy. (all is like the domino effect : a falling row of dominoes).