Interference of two perpendicular standing waves on a string (Melde's Experiment)

Recently, I did Melde's Experiment in our lab. To say about the procedure, we had to find the frequency of a electrically vibrating tuning fork (connected to a mass through string over a pulley) as shown, in two modes transverse and longitudinal. Now, if the standing wave in any mode with a wavelength $\lambda$ , wave velocity $v$ , tension in the string be $T$ and mass per unit length of the string be $\mu$ with the distance between two consecutive nodes as $L$ , then: $\lambda=2L$ and $v=\sqrt{\frac{T}{\mu}}\implies \text{Frequency, }f=\frac{1}{\lambda}\sqrt{\frac{T}{\mu}}\implies f=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$ Now, the frequency of the tuning fork $(N)$ can be determined based on which mode it is being operated upon i.e. N=f \text{, in the transverse mode}\\N=2f \text{, in the longitudinal mode} \tag{1} First, we performed the experiment in transverse mode where the string and prong of tuning fork are parallel, $N$ came out to $66\text{ }Hz$ . But, when we performed in the longitudinal mode , it came out to be $96\text{ }Hz$ i.e. $f$ in this case was averagely around $48\text{ }Hz$ , a drastic error as it turned out that the true predetermined frequency is actually around $65\text{ }Hz$ . Actually, from the starting the longitudinal mode setup was done wrong , since the prong and string wasn't exactly perpendicular, instead it was inclined at an around $30$ degrees with the perpendicular direction. This raised a question that there must be scaling factor between the frequency of the standing wave and of the tuning fork at every angle between the string and the prong as in the $eq(1)$ for the angles $\theta=0$ and $\theta=\pi/2$ . I then tried to find this scaling factor and reached this far as below: Let the angle between the prong and the string be $\theta$ and let the amplitude of oscillation of the tuning fork be $A$ . Now, for a tuning fork vibrating at an amplitude $A$ , amplitude of oscillation of standing wave on string in purely transverse mode $(\theta=0)$ would be $A$ and that in purely longitudinal mode $(\theta=\pi/2)$ can be approximated to be around $A$ for a fork vibrating with high frequency, say $N$ . Then, we can take the components of amplitude $A$ in this mixed state to be $A \cos{\theta}$ and $A \sin{\theta}$ in the transverse and longitudinal directions as shown below: So, there would be two standing waves generated due to each component, perpendicular in directions i.e. one in vertical direction (longitudinal mode) and one in horizontal (transverse mode). Now, we were able to observe completely standing waves i.e. standing nodes in certain cases, hence I am assuming that both standing waves would have the same wavelength in some cases, say $\lambda$ , since $\lambda=2L$ . So, the wave equations of the two waves can be written as: z=(A \cos{\theta}) \sin{(\frac{2\pi x}{\lambda})} \cos{(2\pi Nt)}\tag{2} and y=(A \sin{\theta}) \sin{(\frac{2\pi x}{\lambda})} \cos{(\pi Nt)}\tag{3} where, is the string is assumed to be along the $x$ -axis. The $eq(2)$ and $eq(3)$ represents the transverse and longitudinal waves respectively. Also, the frequency is taken as $N$ and $N/2$ as even after distributing components, the fork would vibrate with the same frequency . Hence, the resultant would be: R=\sqrt{z^2+y^2}=Asin{(\frac{2\pi x}{\lambda})}\sqrt{\cos^2{\theta}\cos^2{(2\pi Nt)}+\sin^2{\theta}\cos^2{(\pi Nt)}}\tag{4} I thought the equation would take a simple form as : $R=Asin{(\frac{2\pi x}{\lambda})}\cos{2\pi f' t}$ where, $f'$ is a function of $N$ . But the above equation doesn't even seem to simplify further. So, how can I simplify it and if it can't be simplified then, is there any wrong assumptions in my analysis which gave this uncomplete answer ? How can this scaling factor be found? It also seems like that if there exists such a factor, it would be time dependent as there is time $t$ in the last equation in the analysis. Any comment on this?