I understand the mathematical origin of complex eigenvalues, and that complex eigenvalues come in pairs. But what is the meaning of the imaginary part? In particular I refer to an acoustic problem (Helmholtz equation) coupled with plate vibrations: the eigenvalues represent frequencies, so what's the physical meaning of two eigenvalues with the same real value and opposite imaginary values (conjugated)? What's the meaning of pure imaginary eigenvalues?
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1$\begingroup$ This is 100% a guess since I don't work in acoustics but if the frequency is complex, then the imaginary part is likely a phase shift. $\endgroup$– tpg2114Oct 14, 2014 at 21:18
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9$\begingroup$ The imaginary part represents damping. If you write a harmonic function with an exponential $e^{i(\omega+i\lambda) t}=e^{i\omega t}e^{-\lambda t}$, then the real exponent $\omega$ represents the frequency and the coefficient of the imaginary part $\lambda$ is the damping constant. Negative $\lambda$ leads to exponentially increasing solutions. $\endgroup$– CuriousOneOct 15, 2014 at 2:07
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2$\begingroup$ @Sparkler: the question is faulty. To ask for a physical sense, you have to specify, eigenvalues of what do you consider (let A be linear operator/matrix of … ). The Laplace operator (from Helmholtz equation) is self-conjugated, hence its eigenvalues are necessarily real, and it isn’t a good candidate to ask about a meaning of imaginary part ☺ $\endgroup$– Incnis MrsiNov 21, 2014 at 13:59
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$\begingroup$ could you please provide reference? $\endgroup$– SparklerNov 21, 2014 at 20:38
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$\begingroup$ @Sparkler: about self-adjointness (note Ī used a non-standard term in the previous comment) of the Laplacian? IMHO any textbook on operator theory that considers this operator at all. You also may search for “Laplacian eigenvalues” in the Internet. $\endgroup$– Incnis MrsiNov 22, 2014 at 13:44
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2 Answers
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To elaborate on one of the comments...
There is no physical meaning to a complex number... until you give it one.
The reason complex numbers appear in the solution to the wave equation, Helmholtz equation, and the harmonic oscillator, is that we continue the field, pressure, or whatever, into the complex plane to make the equation easier to solve. You could avoid this all your life and assume a superposition of real functions and solve for coefficients, etc., as is taught in basic elementary differential equations textbooks.
Using a complex field for the problems makes solving a bit easier but you need to take the real part of your answer.
But in the end when you have damping in a system the "complex eigenvalues" will have a real and imaginary part. If you assume a solution of the form $p = p_0e^{ikx}$ etc, then the real part of k will describe oscillations while the imaginary part of k will describe the damping or attenuation on the field. If you assume a solution of the form $p = p_0e^{kx}$ then the meaning changes. So you see, not only is the "complex" nature dependent on how you go about solving the equation but the "physical meaning" depends on representation.
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What will be interesting about the integrated solution of the nonlinear equation of Navier-Stoks. If in linear equations you can take the real part and you get a real solution, then this is impossible in nonlinear equations. The sum of two decisions is not a decision. If we add the complex and complex conjugate solutions and divide by two, then we will not get a solution to a non-linear equation. In the case of a nonlinear complex solution of an equation, a special formula is needed to convert it into a real solution. We must think about the physical sense of a complex turbulent solution. An imaginary part of a complex turbulent solution means the average quadratic deviation of the real part, which is the average value of the solution. There is a final formula for recalculating a complex turbulent solution into a real $$\frac{d x_k}{d t}=ReV_k(x_1,x_2,x_3)+iImV_k(x_1,x_2,x_3)$$ $$x_k=x_k(t,t_0,x_{10},x_{20},x_{30})$$
In this case, the initial conditions are complex. The complex solution is recalculated into a real one by the formula. But the rms value can have a plus or minus sign, so it must be multiplied by a sine with a variable phase. The oscillation frequency of a turbulent flow is determined by the speed rotor, so I used this formula for the frequency.
$$y_k(t,t_0,x_{10},x_{20},x_{30})=Rex_k(t,t_0,x_{10},x_{20},x_{30})+$$ $$+Imx_k(t,t_0,x_{10},x_{20},x_{30})sin[\int_0^t e_{kpq} \frac{\partial V_q}{\partial x_p}dt+arg(x_{k0})]$$ The result is chaotically oscillating real streamlines with an average equal to the real part.
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2$\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$– Community BotSep 10, 2022 at 18:37