Solutions to Math Of Spherical Harmonics Problems

 1)Suppose the solar p-modes create a surface pattern of nodes and waves such as shown below:

 

Use this diagram to deduce the values for ℓ  and m.  

Solution:

Since ℓ = 10 = m on inspection)  then there are no lines of latitude because these are defined: (ℓ  - m), so (10 – 10) = 0.  Since m = 10 we have an azimuthal order of 10 which indicates 10 nodes (each with embedded contours) around the equator on the three dimensional spherical surface. Five of these are shown and the remaining five can be counted by using the spacing in the sphere foreground.

2)   Explain the appearance of the spherical surface shown below if it describes m= 5 and ℓ = 5. Thence, find the associated Legendre function: P _ℓm (q ) and also Y _ℓm (q,j) assuming q = p/2: (Assume  n _nℓm =  1 /s)

Solution:

Since ℓ = 5 = m then there are no lines of latitude because these are defined: (ℓ  - m), so (5 – 5) = 0.  Since m = 5 we have an azimuthal order of 5 which indicates 5 nodes (each with embedded contours) around the equator on the three dimensional spherical surface. Three of these are shown and the remaining two are on the distal side of the sphere.

Finally we have the angular degree ℓ or the number of nodes along longitude or meridian lines. Again 3 of these (“doubled” longitudes each enfolding a node m) are shown, and 2 others are on the other side of the sphere.     The associated Legendre polynomials are given by:

P _ℓm (q )  =  (1 – z²) ^m/2 / ℓ! 2^ℓ   d ^(ℓ+m) / dz^(ℓ+m)  (z²   -1) ^ℓ

But ℓ  = m = 5 ,  so:

P _ℓm (q )  =  (1 – z²) ⁰ / 5! 2⁵   d⁽⁵⁺⁵⁾  / dz⁽⁵⁺⁵⁾   (z²   -1) ⁵

For q = p/2 ,  d⁽¹⁰⁾  / dz⁽¹⁰⁾   (z²   -1) ⁵= 0  (since z = cos  q)

Therefore: P _ℓm (q )  =  0

Y _ℓm (q,j) =  (1) ^m [2 ℓ +1 (ℓ - m)!/ 4n ((ℓ +m)!] ^½ P ^ℓ_m (z ) exp (i m j),

But since P _ℓm (q )  =  0  is a factor then: Y _ℓm (q,j) =  0