There is no one to one relationship between rise time and bandwidth. A slew rate limiter is a non-linear filter, so can't directly be characterized as a low pass filter with some obvious rolloff frequency. Think of it in the time domain, and you can see that a slew rate limit effects signals proportional to amplitude. A 5 Vpp signal limited to 5 V/µs can't have a period shorter than 2 µs, at which point it degenerates to a 500 kHz triangle wave. However, if the amplitude only needed to be 1 Vpp, then the limit is a 2.5 MHz triangle wave. Since the concept of bandwidth get less clear when a non-linear filter is envolved, you can at best talk about it approximately.
Your answer can also vary greatly depending on what exactly "rise time" is. This is a term that should never be used without some qualification. Even a simple R-C filter has ambiguous rise time. Its step response is a exponential with no place being a clear "end". It's rise time is therefore infinite. Without a threshold of how close to the end you need to be to considered to have risen, the term "rise time" is meaningless. This is why you need to either talk about rise time to a specific fraction of the final value, or slew rate.
The equation you site is therefore just plain wrong, at least without a set of qualifications. Perhaps those are found on the page you got it from, but quoting it out of contect makes it wrong. Your question is unaswerable in its current form.
Added:
You now say the real issue is limiting high frequencies from sharp edges so that parts of the signal don't get into the frequency range where your wire becomes a transmission line. This has little directly to do with rise time. Since the real issue is frequency content, deal with that directly. The simplest way is probably a R-C low pass filter. Set it to roll off above the highest frequency of interest in the signal, and well below the frequency at which your system can no longer be considered lumped. If there is no frequency space between these, then you can't what you want. In that case you need to use a lower bandwidth signal, a shorter wire, or deal with the transmission line aspects of the wire.
In your case, you say the highest frequency of interest is 30 MHz, so adjust the filter to that or a little higher, let's say 50 MHz since that will leave your desired signal pretty much intact. The wavelength of 50 MHz is 6 meters in free space. You didn't say what impedence your transmission line is, but let's figure propagation will be half the speed of light, which leaves 3 meter wavelength on the wire. To be pretty safe just ignoring transmission line issues, you want the wire to be 1/10 wavelength or less, which is 300 mm or about a foot. So if the wire is a foot or less in length, then you can add a simple R-C filter at 50 MHz and forget about it.
Transmission line effects don't just suddenly appear at some magic wavelength relative to the wire length, so how long is too long is a gray area. Up to 1/4 wavelength can often be short enough. If it is "long", then the best thing is to use a impedence controlled driver and a terminator at the other end. However, that is cumbersome and also attenuates the signal by half. You either deal with the lower amplitude at the receiver, or boost it at the transmitter before it gets divided by the driving impedence and the transmission line characteristic impedence.
A simpler solution that may take some experimental tweaking, is to simply put a small resistor in series with the driver and be done with it. That will form a low pass filter with the capacitance of the cable and whatever other stray capacitance is around. It's not as predictable as a deliberate R-C, but much simpler and often good enough.
The question is "is the extra noise you saw relevant to what you are trying to achieve"? In other words is the noise in the bandwidth of interest of your signal? If it isn't, can it still cause you problems? Also, are you sacrificing anything by using the TL051 - if not then use it but there are other considerations.
Op-amps may specify so many nano volts per root hertz but, there is also the L.F. noise (usually specified as so many micro-volts peak-to-peak between 0.1Hz and 10Hz). Was the "so-called" good device actually worse than the TL051 in this area. BTW I haven't looked so I don't know but you shouldn't make assumptions.
The other equally important noise source is due to input bias currents and these are usually in the pico-amps per root hertz range. If you are using large input/feedback resistors this noise can dominate and, you might find that the TL051 is significantly better than the AD829 in this area. Again, I haven't looked but it's always best not to make assumptions.
EDIT - the TL051 is 150x better than the AD829 for input current noise. TL051 is 0.01pA per sqrt(Hz) whilst the AD829 is 1.5pA per sqrt(Hz). This will make a significant difference if the size of resistors on inputs and feedback are 1kohm or above. At 1kohm, the AD829's input current noise becomes about the same level as its input voltage noise and at 10kohm it becomes comparable with the TL051's voltage noise.
Power supply rejection ratio is another thing to consider - if there is noise on the power supply (as there inevitably is), which device is better at rejecting it?
Common mode rejection - if the amp configuration is differential, you would naturally assume common-mode noise would be always near zero - not so, check the data sheets and you will see that C.M.R. ratio gets far worse at higher frequencies.
I'm just trying to say that it's always a more complex picture!!!
Best Answer
First to your question: The term "Average received signal power over bandwidth" is the power spectral density (PSD), usually in unit dBm/Hz or W/Hz. It cannot be used directly to calculate channel capacity, it's prime use is to calculate interference.
For discussion of channel capacity, it is important we are not dealing with "distortion" (which would be counteracted by pre-distortion and line coding), but with additional noise. There are different sources of noise with different spectra, but for discussion of channel coding it is sufficient to look at additional white gaussian noise (AWGN), a memoryless stochastic process totally uncorrelated to your signal.
AWGN has in infinite bandwidth, but limited radiant intensity. This opens the door to reduction of total received noise power while keeping the full total received signal power, thus improving SNR.
At each reference point in your receiver, you are able to divide total signal by noise power and tell a signal to noise ratio (SNR), typically in decibels. Important for channel capacity is the SNR before channel decoding. Now Shannon comes into play.
He states that information can be transferred as long as the SNR (in dB) is \$>-\infty\$. A common misunderstanding is that SNR > 0 is required, but this is just the point where detection and demodulation of the signal become very easy. Detection is harder for SNR < 0 but still any deviation from a pure stochastic process is detectable. The reasoning that information transfer with arbitrary low bit error rate is possible for every SNR is quite surprising and makes Shannons original article an interesting read.
The rate at which information can be transferred is however limited and Shannon is able to proove an upper bound for this rate. It depends on occupied bandwidth, which is related to maximum slew rate (think of Fourier decomposition) and minimum temporal spacing of symbols, and the SNR, requiring a minimum symbol distinctness and limiting bits per symbol.
The Shannon-Limit is a theoretical upper bound, channel codes with near maximum performance are still an active field of research. LPDC-Codes and Turbo-Codes are the best known solutions.