Problems 311
PROBLEMS
14.1 Start with expressions (1.33) and (1.34) and derive the directivity pattern (14.4) of acoustic waves scattered at a soft cylinder. See details in Sections 6.2 and 13.1.1.
14.2 Start with expressions (1.33) and (1.34) and derive the directivity pattern (14.5) of acoustic waves scattered at a hard cylinder. See details in Sections 6.2 and 13.1.1.
14.3 Use Equations (1.53) and (14.4) and derive the total cross-section (14.7) of a soft cylinder.
14.4 Use Equations (1.53) and (14.5) and derive the total cross-section (14.7) of a hard cylinder.
14.5 Use expression (14.4) and compute the normalized scattering cross-sections (14.9) for the individual parts of a soft cylinder with diameter d = 2a = λ and length L = 2l = 3λ. Set γ = 45◦. Represent the results in the form of a diagram as in Figure 14.2. Write the expression for the electromagnetic wave incident on a perfectly conducting cylinder, whose PO scattering cross-section is identical to that plotted by you.
14.6 Use expression (14.5) and compute the normalized scattering cross-sections (14.9) for the individual parts of a hard cylinder with diameter d = 2a = λ and length L = 2l = 3λ. Set γ = 45◦. Represent the results in the form of a diagram as in Figure 14.4. Write the expression for the electromagnetic wave incident on a perfectly conducting cylinder, whose PO scattering cross-section is identical to that plotted by you.
14.7 Use Equations (14.4) and (14.5) and compute the directivity pattern (14.9) of the shadow radiation (14.10) for the cylinder with diameter d = 2a = λ and length L = 2l = 3λ. Set γ = 45◦. Represent the results in the form of a diagram as in Figure 14.6. Write the expression for the electromagnetic wave incident on a perfectly conducting cylinder and generating the same shadow radiation as that plotted by you.
14.8 The incident wave (14.1) hits a hard cylinder (Fig. 14.1). Use the asymptotic expression (7.90) and derive the function (14.12), which describes the field radiated by the scattering sources jh(1) induced near the left edge. Write the explicit expressions for the function Fh(1) in terms of functions Vt (σ1, ϕ0), V0(β1, ϕ0), Vt (σ2, α − ϕ0), V0(β2, α − ϕ0) and define the parameters σ1,2, β1,2.
14.9 Elementary edge waves are the functions of the local angles γ0, θ . Derive Equations
ˆ ˆ
(14.14), which define these angles. Use the dot products between the unit vectors R, t
ˆ ˆi ˆ ˆ
and t, k where R shows the scattering direction in the plane y0z, t is the tangent to the
ˆi shows the direction of the incident wave. edge, and k
14.10Elementary edge waves are the functions of the local angles φ, φ0. Derive Equations (14.15) and (14.16), which define these angles. Hint: project the unit vectors
|
|
= ˆ |
+ ˆ |
= − |
ki |
= −ˆ |
− ˆ |
|
Rˆ |
ˆ |
|
y sin |
z cos and Qˆ |
|
y sin γ |
z cos γ on the plane normal to the |
tangent ˆt. Recall the note box following Equation (7.131).
14.11Apply the stationary-phase technique to the second terms in Equations (14.4) and (14.5), and derive asymptotics (14.22), (14.23) for the beam scattered from the lateral/cylindrical surface.
14.12Use Equation (14.12), apply the stationary-phase technique, and derive the asymptotic expression (similar to Equation (14.31)) for the edge-diffracted ray diverging from the stationary point 1 (Fig. 14.1).
14.13Verify that the incident wave (13.42) generates the surface current (13.43) on the left base (disk) of a perfectly conducting cylinder (Fig. 13.1). Use Equations (1.91), (1.92),