Expressing $Asin(kx-wt) + Bsin(kx+wt+d)$ in the form $C sin(cdot)$ [on hold]

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Does someone know the expression for $Asin(kx-wt) + Bsin(kx+wt+d)$ off the top of their head? $k$, $w$, $A$, $B$, and $d$ are constants. I would like to write the expression as $Csin(cdot)$, where $C$ doesn't contain $x$ or $t$.

trigonometry

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edited Aug 3 at 3:28

Blue

43.6k868141

asked Aug 3 at 3:04

fibo11235

365

put on hold as off-topic by Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:55

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
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  Is there a context to your question? That may be helpful in figuring out what exactly your trying to do and if there are any easier ways to solving your problem.
  – Aaron Quitta
  Aug 3 at 3:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It is the superposition of two waves traveling in opposite directions.
  – fibo11235
  Aug 3 at 3:38
  

  
  
  
  


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  1
  

  
  
  
  
  
  

  
  If $B$ were equal to $A$, the expression could be written as $$2 A cosleft(wt+fracd2right) sinleft(kx+fracd2right)$$ This doesn't seem to be the form you want. In full generality, the expression certainly doesn't reduce to the form you want.
  – Blue
  Aug 3 at 3:40
  
  

  
  
  
  


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  How about if the +wt in the second expression is also a minus?
  – fibo11235
  Aug 3 at 3:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @fibo11235 -- Then the two waves are moving in the same direction. It still doesn't simplify if $Aneq B$; a general sum of sine waves is not a sine wave. See Fourier series.
  – mr_e_man
  Aug 3 at 3:47
  
  

  
  
  
  

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up vote
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favorite

Does someone know the expression for $Asin(kx-wt) + Bsin(kx+wt+d)$ off the top of their head? $k$, $w$, $A$, $B$, and $d$ are constants. I would like to write the expression as $Csin(cdot)$, where $C$ doesn't contain $x$ or $t$.

trigonometry

share|cite|improve this question

edited Aug 3 at 3:28

Blue

43.6k868141

asked Aug 3 at 3:04

fibo11235

365

put on hold as off-topic by Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:55

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Is there a context to your question? That may be helpful in figuring out what exactly your trying to do and if there are any easier ways to solving your problem.
  – Aaron Quitta
  Aug 3 at 3:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It is the superposition of two waves traveling in opposite directions.
  – fibo11235
  Aug 3 at 3:38
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  If $B$ were equal to $A$, the expression could be written as $$2 A cosleft(wt+fracd2right) sinleft(kx+fracd2right)$$ This doesn't seem to be the form you want. In full generality, the expression certainly doesn't reduce to the form you want.
  – Blue
  Aug 3 at 3:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  How about if the +wt in the second expression is also a minus?
  – fibo11235
  Aug 3 at 3:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @fibo11235 -- Then the two waves are moving in the same direction. It still doesn't simplify if $Aneq B$; a general sum of sine waves is not a sine wave. See Fourier series.
  – mr_e_man
  Aug 3 at 3:47
  
  

  
  
  
  

 | 
show 1 more comment

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Does someone know the expression for $Asin(kx-wt) + Bsin(kx+wt+d)$ off the top of their head? $k$, $w$, $A$, $B$, and $d$ are constants. I would like to write the expression as $Csin(cdot)$, where $C$ doesn't contain $x$ or $t$.

trigonometry

share|cite|improve this question

edited Aug 3 at 3:28

Blue

43.6k868141

asked Aug 3 at 3:04

fibo11235

365

Does someone know the expression for $Asin(kx-wt) + Bsin(kx+wt+d)$ off the top of their head? $k$, $w$, $A$, $B$, and $d$ are constants. I would like to write the expression as $Csin(cdot)$, where $C$ doesn't contain $x$ or $t$.

trigonometry

share|cite|improve this question

edited Aug 3 at 3:28

Blue

43.6k868141

asked Aug 3 at 3:04

fibo11235

365

share|cite|improve this question

share|cite|improve this question

edited Aug 3 at 3:28

Blue

43.6k868141

edited Aug 3 at 3:28

Blue

43.6k868141

edited Aug 3 at 3:28

Blue

43.6k868141

43.6k868141

asked Aug 3 at 3:04

fibo11235

365

asked Aug 3 at 3:04

fibo11235

365

asked Aug 3 at 3:04

fibo11235

365

365

put on hold as off-topic by Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:55

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz

If this question can be reworded to fit the rules in the help center, please edit the question.

put on hold as off-topic by Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz Aug 3 at 7:55

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Isaac Browne, Piyush Divyanakar, Henrik, max_zorn, Mostafa Ayaz

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Is there a context to your question? That may be helpful in figuring out what exactly your trying to do and if there are any easier ways to solving your problem.
  – Aaron Quitta
  Aug 3 at 3:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It is the superposition of two waves traveling in opposite directions.
  – fibo11235
  Aug 3 at 3:38
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  If $B$ were equal to $A$, the expression could be written as $$2 A cosleft(wt+fracd2right) sinleft(kx+fracd2right)$$ This doesn't seem to be the form you want. In full generality, the expression certainly doesn't reduce to the form you want.
  – Blue
  Aug 3 at 3:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  How about if the +wt in the second expression is also a minus?
  – fibo11235
  Aug 3 at 3:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @fibo11235 -- Then the two waves are moving in the same direction. It still doesn't simplify if $Aneq B$; a general sum of sine waves is not a sine wave. See Fourier series.
  – mr_e_man
  Aug 3 at 3:47
  
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Is there a context to your question? That may be helpful in figuring out what exactly your trying to do and if there are any easier ways to solving your problem.
  – Aaron Quitta
  Aug 3 at 3:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It is the superposition of two waves traveling in opposite directions.
  – fibo11235
  Aug 3 at 3:38
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  If $B$ were equal to $A$, the expression could be written as $$2 A cosleft(wt+fracd2right) sinleft(kx+fracd2right)$$ This doesn't seem to be the form you want. In full generality, the expression certainly doesn't reduce to the form you want.
  – Blue
  Aug 3 at 3:40
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  How about if the +wt in the second expression is also a minus?
  – fibo11235
  Aug 3 at 3:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @fibo11235 -- Then the two waves are moving in the same direction. It still doesn't simplify if $Aneq B$; a general sum of sine waves is not a sine wave. See Fourier series.
  – mr_e_man
  Aug 3 at 3:47
  
  

  
  
  
  

1

1

Is there a context to your question? That may be helpful in figuring out what exactly your trying to do and if there are any easier ways to solving your problem.
– Aaron Quitta
Aug 3 at 3:27

Is there a context to your question? That may be helpful in figuring out what exactly your trying to do and if there are any easier ways to solving your problem.
– Aaron Quitta
Aug 3 at 3:27

It is the superposition of two waves traveling in opposite directions.
– fibo11235
Aug 3 at 3:38

It is the superposition of two waves traveling in opposite directions.
– fibo11235
Aug 3 at 3:38

1

1

If $B$ were equal to $A$, the expression could be written as $$2 A cosleft(wt+fracd2right) sinleft(kx+fracd2right)$$ This doesn't seem to be the form you want. In full generality, the expression certainly doesn't reduce to the form you want.
– Blue
Aug 3 at 3:40

If $B$ were equal to $A$, the expression could be written as $$2 A cosleft(wt+fracd2right) sinleft(kx+fracd2right)$$ This doesn't seem to be the form you want. In full generality, the expression certainly doesn't reduce to the form you want.
– Blue
Aug 3 at 3:40

How about if the +wt in the second expression is also a minus?
– fibo11235
Aug 3 at 3:45

How about if the +wt in the second expression is also a minus?
– fibo11235
Aug 3 at 3:45

1

1

@fibo11235 -- Then the two waves are moving in the same direction. It still doesn't simplify if $Aneq B$; a general sum of sine waves is not a sine wave. See Fourier series.
– mr_e_man
Aug 3 at 3:47

@fibo11235 -- Then the two waves are moving in the same direction. It still doesn't simplify if $Aneq B$; a general sum of sine waves is not a sine wave. See Fourier series.
– mr_e_man
Aug 3 at 3:47

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1 Answer
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In the comments, the original poster asked what would happen if the $-$ became a $+$ in the second equation, meaning both waves were travelling in the same direction. In this case the problem is expressive as desired, in the case of opposite waves I am still considering.

The Harmonic Function Theorem states that for the addition of two general sine waves, $psi_1=Asin(wt+delta_1)$, $psi_2=Bsin(wt+delta_2)$, their sum is expressible as $C sin(wt+delta)$. In the final term, $C=sqrtA^2+B^2+2ABcos(delta_2-delta_1)$, and $delta=arctanleft(fracAsin(delta_1)+Bsin(delta_2)Acos(delta_1)+Bcos(delta_2)right)$.

Applying this to your problem, we have $Asin(kx+wt)+Bsin(kx+wt+d)$. Directly applying the theorem, we have $C=sqrtA^2+B^2+2ABcos(d)$ (note: this is the reason that in order to solve this without $C$ containing any $t$, it is important that they are both going in the same direction. If they were going opposite, they wouldn’t have cancelled, and the argument of the cosine would’ve been ($d+2t$)). As for $delta$, we have $arctan left(fracAsin(wt)+Bsin(wt+d)Acos(wt)+Bcos(wt+d)right)$

Therefore, it is possible to express the sum as $Csin(wt+delta)$, with $C$ and $delta$ defines above.

Note: the choice of whether to include $wt$ or $kx$ in the answer is arbitrary. One can choose to isolate either and include the other in the $delta$, it results in the same wave.

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answered Aug 3 at 7:08

Tyler6

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1 Answer
1

active

oldest

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1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

In the comments, the original poster asked what would happen if the $-$ became a $+$ in the second equation, meaning both waves were travelling in the same direction. In this case the problem is expressive as desired, in the case of opposite waves I am still considering.

The Harmonic Function Theorem states that for the addition of two general sine waves, $psi_1=Asin(wt+delta_1)$, $psi_2=Bsin(wt+delta_2)$, their sum is expressible as $C sin(wt+delta)$. In the final term, $C=sqrtA^2+B^2+2ABcos(delta_2-delta_1)$, and $delta=arctanleft(fracAsin(delta_1)+Bsin(delta_2)Acos(delta_1)+Bcos(delta_2)right)$.

Applying this to your problem, we have $Asin(kx+wt)+Bsin(kx+wt+d)$. Directly applying the theorem, we have $C=sqrtA^2+B^2+2ABcos(d)$ (note: this is the reason that in order to solve this without $C$ containing any $t$, it is important that they are both going in the same direction. If they were going opposite, they wouldn’t have cancelled, and the argument of the cosine would’ve been ($d+2t$)). As for $delta$, we have $arctan left(fracAsin(wt)+Bsin(wt+d)Acos(wt)+Bcos(wt+d)right)$

Therefore, it is possible to express the sum as $Csin(wt+delta)$, with $C$ and $delta$ defines above.

Note: the choice of whether to include $wt$ or $kx$ in the answer is arbitrary. One can choose to isolate either and include the other in the $delta$, it results in the same wave.

share|cite|improve this answer

answered Aug 3 at 7:08

Tyler6

440210

add a comment | 

up vote
0
down vote

In the comments, the original poster asked what would happen if the $-$ became a $+$ in the second equation, meaning both waves were travelling in the same direction. In this case the problem is expressive as desired, in the case of opposite waves I am still considering.

The Harmonic Function Theorem states that for the addition of two general sine waves, $psi_1=Asin(wt+delta_1)$, $psi_2=Bsin(wt+delta_2)$, their sum is expressible as $C sin(wt+delta)$. In the final term, $C=sqrtA^2+B^2+2ABcos(delta_2-delta_1)$, and $delta=arctanleft(fracAsin(delta_1)+Bsin(delta_2)Acos(delta_1)+Bcos(delta_2)right)$.

Applying this to your problem, we have $Asin(kx+wt)+Bsin(kx+wt+d)$. Directly applying the theorem, we have $C=sqrtA^2+B^2+2ABcos(d)$ (note: this is the reason that in order to solve this without $C$ containing any $t$, it is important that they are both going in the same direction. If they were going opposite, they wouldn’t have cancelled, and the argument of the cosine would’ve been ($d+2t$)). As for $delta$, we have $arctan left(fracAsin(wt)+Bsin(wt+d)Acos(wt)+Bcos(wt+d)right)$

Therefore, it is possible to express the sum as $Csin(wt+delta)$, with $C$ and $delta$ defines above.

Note: the choice of whether to include $wt$ or $kx$ in the answer is arbitrary. One can choose to isolate either and include the other in the $delta$, it results in the same wave.

share|cite|improve this answer

answered Aug 3 at 7:08

Tyler6

440210

add a comment | 

up vote
0
down vote

up vote
0
down vote

In the comments, the original poster asked what would happen if the $-$ became a $+$ in the second equation, meaning both waves were travelling in the same direction. In this case the problem is expressive as desired, in the case of opposite waves I am still considering.

The Harmonic Function Theorem states that for the addition of two general sine waves, $psi_1=Asin(wt+delta_1)$, $psi_2=Bsin(wt+delta_2)$, their sum is expressible as $C sin(wt+delta)$. In the final term, $C=sqrtA^2+B^2+2ABcos(delta_2-delta_1)$, and $delta=arctanleft(fracAsin(delta_1)+Bsin(delta_2)Acos(delta_1)+Bcos(delta_2)right)$.

Applying this to your problem, we have $Asin(kx+wt)+Bsin(kx+wt+d)$. Directly applying the theorem, we have $C=sqrtA^2+B^2+2ABcos(d)$ (note: this is the reason that in order to solve this without $C$ containing any $t$, it is important that they are both going in the same direction. If they were going opposite, they wouldn’t have cancelled, and the argument of the cosine would’ve been ($d+2t$)). As for $delta$, we have $arctan left(fracAsin(wt)+Bsin(wt+d)Acos(wt)+Bcos(wt+d)right)$

Therefore, it is possible to express the sum as $Csin(wt+delta)$, with $C$ and $delta$ defines above.

Note: the choice of whether to include $wt$ or $kx$ in the answer is arbitrary. One can choose to isolate either and include the other in the $delta$, it results in the same wave.

share|cite|improve this answer

answered Aug 3 at 7:08

Tyler6

440210

In the comments, the original poster asked what would happen if the $-$ became a $+$ in the second equation, meaning both waves were travelling in the same direction. In this case the problem is expressive as desired, in the case of opposite waves I am still considering.

The Harmonic Function Theorem states that for the addition of two general sine waves, $psi_1=Asin(wt+delta_1)$, $psi_2=Bsin(wt+delta_2)$, their sum is expressible as $C sin(wt+delta)$. In the final term, $C=sqrtA^2+B^2+2ABcos(delta_2-delta_1)$, and $delta=arctanleft(fracAsin(delta_1)+Bsin(delta_2)Acos(delta_1)+Bcos(delta_2)right)$.

Applying this to your problem, we have $Asin(kx+wt)+Bsin(kx+wt+d)$. Directly applying the theorem, we have $C=sqrtA^2+B^2+2ABcos(d)$ (note: this is the reason that in order to solve this without $C$ containing any $t$, it is important that they are both going in the same direction. If they were going opposite, they wouldn’t have cancelled, and the argument of the cosine would’ve been ($d+2t$)). As for $delta$, we have $arctan left(fracAsin(wt)+Bsin(wt+d)Acos(wt)+Bcos(wt+d)right)$

Therefore, it is possible to express the sum as $Csin(wt+delta)$, with $C$ and $delta$ defines above.

Note: the choice of whether to include $wt$ or $kx$ in the answer is arbitrary. One can choose to isolate either and include the other in the $delta$, it results in the same wave.

share|cite|improve this answer

answered Aug 3 at 7:08

Tyler6

440210

share|cite|improve this answer

share|cite|improve this answer

answered Aug 3 at 7:08

Tyler6

440210

answered Aug 3 at 7:08

Tyler6

440210

answered Aug 3 at 7:08

Tyler6

440210

440210

add a comment | 

add a comment |