4) Normal mode analysis. Assume a linear wave that varies as [i(𝐤·𝐱-ω t)]. This allows you to re

In this question we have given a uniform string of length 20 meter and it is suspended from a ri

In this question we are given u of x t k is equal to exponential of i times k x minus omega as a

Okay, so in this problem we will be showing that that k square is equal to omega square over c s

Hello friends in this question the solution for the p .d is given as u of x comma t as equal to

Let's discuss this question. So here one dimensional wave equation is given to us and we need to

Question

4) Normal mode analysis. Assume a linear wave that varies as \( \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)] \). This allows you to replace time derivatives with \( -i \omega \), spatial derivatives with \( i \mathbf{k} \), and first order quantities with \( a_{1} \longrightarrow \tilde{a} \). Rewrite the five equations of part 3 using these substitutions. (Hint: for a one dimensional wave, \( \mathbf{k}=k \).) 5) Dispersion relation. To solve these five equations, eliminate two of them to write the three remaining equations in the form \[ \left(\begin{array}{ccc} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{array}\right)\left(\begin{array}{c} \tilde{\phi} \\ \tilde{n}_{i} \\ \tilde{u}_{i} \end{array}\right)=\left(\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right) \] The determinant of this equation (which you do not have to solve) yields the following 2 dispersion relation: \[ \omega^{2}=k^{2}\left[\frac{Z_{i} T_{e}}{m_{i}\left(1+k^{2} \lambda_{D e}^{2}\right)}+\frac{\gamma T_{i}}{m_{i}}\right] \]

          4) Normal mode analysis. Assume a linear wave that varies as \( \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)] \). This allows you to replace time derivatives with \( -i \omega \), spatial derivatives with \( i \mathbf{k} \), and first order quantities with \( a_{1} \longrightarrow \tilde{a} \). Rewrite the five equations of part 3 using these substitutions. (Hint: for a one dimensional wave, \( \mathbf{k}=k \).)
5) Dispersion relation. To solve these five equations, eliminate two of them to write the three remaining equations in the form
\[
\left(\begin{array}{ccc}
? & ? & ? \\
? & ? & ? \\
? & ? & ?
\end{array}\right)\left(\begin{array}{c}
\tilde{\phi} \\
\tilde{n}_{i} \\
\tilde{u}_{i}
\end{array}\right)=\left(\begin{array}{c}
0 \\
0 \\
0
\end{array}\right)
\]

The determinant of this equation (which you do not have to solve) yields the following
2
dispersion relation:
\[
\omega^{2}=k^{2}\left[\frac{Z_{i} T_{e}}{m_{i}\left(1+k^{2} \lambda_{D e}^{2}\right)}+\frac{\gamma T_{i}}{m_{i}}\right]
\]

4) Normal mode analysis. Assume a linear wave that varies as [i(𝐤·𝐱-ω t)]. This allows you to replace time derivatives with -i ω, spatial derivatives with i 𝐤, and first order quantities with a1⟶ã. Rewrite the five equations of part 3 using these substitutions. (Hint: for a one dimensional wave, 𝐤=k.)
5) Dispersion relation. To solve these five equations, eliminate two of them to write the three remaining equations in the form

(
? ? ?

? ? ?

? ? ?
)(
ϕ̃
ñi
ũi
)=(
0

0

0
)

The determinant of this equation (which you do not have to solve) yields the following
2
dispersion relation:

ω^2=k^2[(Zi Te)/(mi(1+k^2λD e^2))+(γ Ti)/(mi)]

Added by John D.

Question

Please give Ace some feedback