5.16: Time Average of Total Forces and Torques in the Sinusoidal Steady State

Two descriptions are used to generalize the complex amplitude representations describing the stestate response to a sinusoidal drive having the angular frequency \(\omega\). If the system is spatially periodic, or can be modeled by a portion of a periodic system, a Fourier series generalization of the complex amplitude description is appropriate. If it extends to "infinity," a Fourier transform is conveniently made the complex amplitude. The conventions and formulas for computing the time-average of field products, for example of forces, are summarized in this section.

 

Fourier Series Complex Amplitudes

With a periodicity length \(l\) in the \(z\) direction, the Fourier series becomes one of complex amplitudes:

\[ A(z,t) = Re \hat{A} (z, \omega) e^{j \omega t}; \, \hat{A} = \Sigma_{n = - \infty}^{+ \infty} \hat{A}_n (k_n, \omega) e^{-j k_n z} \label{1} \]

where \(k_n \equiv 2n \pi/l\). The series, which determines the phase as well as amplitude of the field at any given point, is in general complex. Thus, \(\hat{A}_n\) is not necessarily equal to \(\hat{A}{-n}^{*}\). Each term in the series can be regarded as a traveling wave with phase velocity \(\omega/k_n\). The Fourier amplitudes are determined by multiplying both sides of Eq. \ref{1} b by \(exp(jk_mz)\), integrating both sides over the length \(l\) and exploiting the orthogonality to solve for \(\hat{A}_m\).With \(m \rightarrow n\),

\[ \hat{A}_n = \frac{1}{l} \int_o^l \hat{A} \, e^{jk_n z} dz \label{2} \]

The time-average of a product of fields \(A\) and \(B\), written in this form, is obtained by regarding each series as the complex amplitude (Eq. 2.15.14, with \(k \rightarrow \omega\) and \(z \rightarrow t\)) to obtain

\[ \langle AB \rangle_t = \frac{1}{2} Re \Big [ \Sigma_{n = - \infty}^{+ \infty} \, \hat{A}_n e^{-jk_n z} \, \Sigma_{m = -\infty}^{+ \infty} \, \hat{B}_m^{*} e^{jk_mz} \Big ] \label{3} \]

The total time-average force (or some other physical quantity involving the product \(AB\)) is the space average of Eq. \ref{3} multiplied by the length. To compute the space-average of the time average, think of writing out the first series in Eq. \ref{3}, and then successively multiplying it by each term from the second series and averaging over the length. Each term from the second series forms only one product having a finite integral over the length \(l\), the term with \(m = n\). Thus, Eq. \ref{3} becomes

\[ \frac{1}{l} \int_{z}^{z+l} \langle AB \rangle_t dz = \frac{1}{2} Re \, \Sigma_{n = -\infty}^{+ \infty} \hat{A}_n \hat{B}_n^{*} \label{4} \]

Application of this expression is illustrated in Sec. 6.4. Its role with respect to Fourier series complex amplitudes is analogous to that of the formula developed next in connection with Fourier transform complex amplitudes.

 

Fourier Transform Complex Amplitudes

In a spatial transient situation, such as illustrated in Sec. 5.17, the complex amplitude takes the form of a Fourier superposition integral:

\[ A(z,t) = Re \hat{A} (z, \omega) e^{j \omega t}; \, \hat{A} = \frac{1}{2 \pi} \int_{- \infty}^{+ \infty} \hat{A} e^{-jkz} dk \label{5} \]

The Fourier transform is found from the complementary integral

\[ \hat{A} = \int_{- \infty}^{+ \infty} \hat{A} e^{jkz} dz \label{6} \]